Chapter 3 — ପରମାଣୁ ଓ ଅଣୁ

By the eighteenth century chemists had learnt to weigh substances carefully before and after a reaction. Antoine Lavoisier, the French scientist, did so with great precision and found a rule that has never been broken: in a chemical reaction the total mass of the products is equal to the total mass of the reactants. This is the law of conservation of mass: mass can neither be created nor destroyed in a chemical reaction.

The law can be verified in the laboratory. Take a solution of about 5.6 g of copper sulphate in 10 mL of water in a conical flask, and a solution of about 1.2 g of sodium carbonate in 10 mL of water in a small ignition tube. Hang the ignition tube inside the flask by a thread so that the solutions do not mix, and close the flask with a cork. Weigh the whole apparatus. Now tilt the flask so that the two solutions mix; a reaction takes place and a greenish precipitate appears. Weigh the apparatus again. The mass is found to be exactly the same. The same result is obtained with solutions of barium chloride and sodium sulphate (white precipitate), or lead nitrate and sodium chloride. The atoms have rearranged themselves into new compounds but not one has been lost or made.

Why must the flask be closed? Because if a reaction produces a gas, as when calcium carbonate reacts with hydrochloric acid to give carbon dioxide, the gas would escape and the mass would appear to decrease; and when magnesium burns, it appears to gain mass, because oxygen from the air has combined with it. In a closed container both cases show conservation. The law thus states that in a chemical reaction the total mass of all the substances present before and after is the same; it is a bookkeeping law, and the reason will become clear from Dalton's theory: the same atoms are present before and after, only differently combined.

An example of the law in numbers: when 12 g of carbon burns completely in 32 g of oxygen, exactly 44 g of carbon dioxide is formed; 12 + 32 = 44. When 4 g of hydrogen burns in 32 g of oxygen, 36 g of water is formed. If a problem gives the masses of all reactants but one product, the missing mass is found by subtraction.

A caution for later study: in nuclear reactions, which are not chemical reactions, a small amount of mass is converted into energy according to Einstein's relation; for the chemical reactions of this course the law holds exactly. The law of conservation of mass is the reason every chemical equation must be balanced — the same number of atoms of each element must appear on both sides.

Lavoisier and his contemporaries also analysed compounds — broke them into their elements and weighed the parts — and the French chemist Joseph Proust, in 1799, stated the second law of chemical combination: in a chemical compound the elements are always present in a definite proportion by mass. This is the law of constant proportions, also called the law of definite proportions.

The standard illustration is water. Whatever its source — the Mahanadi, a well in Koraput, rain over Puri, snow in the Himalaya, or water made in the laboratory by burning hydrogen — pure water always contains hydrogen and oxygen in the ratio 1 : 8 by mass. Decompose 9 g of water and you get 1 g of hydrogen and 8 g of oxygen; decompose 18 g and you get 2 g and 16 g. Likewise, ammonia always contains nitrogen and hydrogen in the ratio 14 : 3 by mass, carbon dioxide has carbon and oxygen in the ratio 3 : 8, and the iron sulphide of Chapter 2 has iron and sulphur in the ratio 7 : 4.

The law has an important consequence: a compound has a fixed composition regardless of how it is made or where it is found. This is precisely what distinguishes a compound from a mixture, whose composition can vary. If more of one element is supplied than the fixed ratio requires, the excess remains uncombined: 3 g of hydrogen and 8 g of oxygen give 9 g of water and 2 g of hydrogen is left over.

Worked example. Hydrogen and oxygen combine in the ratio 1 : 8 by mass to form water. What mass of oxygen is required to react completely with 3 g of hydrogen? Since 1 g of hydrogen needs 8 g of oxygen, 3 g of hydrogen needs 3 × 8 = 24 g of oxygen.

Worked example. In an experiment 1.375 g of copper oxide on reduction gave 1.098 g of copper; in another experiment 1.179 g of copper dissolved in nitric acid and converted to oxide gave 1.476 g of copper oxide. Show that the results agree with the law. In the first, copper : oxygen = 1.098 : (1.375 − 1.098) = 1.098 : 0.277 = 3.96 : 1. In the second, copper : oxygen = 1.179 : (1.476 − 1.179) = 1.179 : 0.297 = 3.97 : 1. The ratio is the same within experimental error, so the law is verified.

Together the two laws — conservation of mass and constant proportions — were facts without an explanation until John Dalton, in 1808, proposed that matter is made of atoms and that compounds are formed by atoms joining in fixed numbers. The next topic sets out his theory and shows how it makes both laws obvious.

The idea that matter is made of indivisible particles is ancient — Kanad in India called the particle parmanu and Democritus in Greece called it atomos, the uncuttable — but it was a philosophy, not a science, until the English schoolteacher and chemist John Dalton turned it into a theory that explained the laws of chemical combination. In 1808 he published his atomic theory, whose postulates are:

1. All matter is made of very tiny particles called atoms. 2. Atoms are indivisible particles that can neither be created nor destroyed in a chemical reaction. 3. Atoms of a given element are identical in mass and chemical properties. 4. Atoms of different elements have different masses and chemical properties. 5. Atoms combine in the ratio of small whole numbers to form compounds. 6. The relative number and kinds of atoms are constant in a given compound.

The theory explains the two laws at once. The law of conservation of mass follows from postulate 2: since atoms are neither created nor destroyed in a reaction but only rearranged, and each atom keeps its mass, the total mass cannot change. The law of constant proportions follows from postulates 5 and 6: a compound is formed by atoms of different elements combining in a fixed whole-number ratio, and since each kind of atom has a fixed mass, the mass ratio of the elements in the compound is fixed too. Water always has two hydrogen atoms for every oxygen atom, and since an oxygen atom is sixteen times as heavy as a hydrogen atom, the mass ratio is 2 : 16, that is 1 : 8.

Dalton was the first scientist to use symbols for elements in a specific sense — his symbols were circles with marks inside — and he used them to represent a definite quantity of the element, one atom, so that a symbol meant both the name and a fixed mass. Later chemists replaced his pictures with letters, as the next topic describes.

Modern science has modified some of Dalton's postulates. The atom is not indivisible: it is made of electrons, protons and neutrons, and can be split in nuclear reactions — though in chemical reactions it does remain intact, which is what Dalton was concerned with. Atoms of the same element are not always identical in mass: isotopes, such as carbon-12 and carbon-14, have different masses. Atoms of different elements can have the same mass number (isobars). And some compounds do not have a simple whole-number ratio. These corrections are taught in the next chapter and in higher classes; they refine the theory but do not overturn it. Dalton's central insight — that chemistry is the combination and rearrangement of atoms in fixed numbers — remains the foundation of the subject, and the law of conservation of mass and the law of constant proportions are, in the atomic view, simply consequences of the way atoms behave.

An atom is the smallest particle of an element that can take part in a chemical reaction. It is the building block of all matter, just as a brick is of a wall — with the difference that there are 118 kinds of brick, one for each element. Atoms of most elements do not exist independently; they join to form molecules or ions, as later topics describe.

How big is an atom? Unimaginably small. The radius of an atom is measured in nanometres (1 nm = 10⁻⁹ m): a hydrogen atom has a radius of about 10⁻¹⁰ m, that is 0.1 nm; a molecule of water is about 10⁻⁹ m; a grain of sand is about 10⁻⁴ m; a plant cell is about 10⁻⁵ m; and a water melon is about 10⁻¹ m. Two hundred million hydrogen atoms placed in a row would just cover the full stop at the end of this sentence. Atoms cannot be seen with the eye or an ordinary microscope, but scanning tunnelling microscopes now produce images of the surfaces of elements in which individual atoms appear as bumps.

Symbols. Dalton used circles with marks. In 1813 the Swedish chemist J. J. Berzelius proposed that each element be given a symbol made from a letter or two of its name, and this is the system in use today; the International Union of Pure and Applied Chemistry (IUPAC) approves the names and symbols of elements. The rules are: the symbol is the first letter of the English name written in capital — H for hydrogen, O for oxygen, C for carbon, N for nitrogen, S for sulphur, P for phosphorus, B for boron, F for fluorine, I for iodine, K for potassium (from its Latin name kalium), U for uranium. When two elements begin with the same letter, the symbol has two letters, the first capital and the second small — Cl for chlorine, Ca for calcium, Co for cobalt, Cu for copper, Cr for chromium, Al for aluminium, Ar for argon, Ba for barium, Br for bromine, Mg for magnesium, Mn for manganese, Ne for neon, Ni for nickel, Si for silicon, Zn for zinc. Some symbols come from Latin, Greek or German names: Fe for iron (ferrum), Na for sodium (natrium), K for potassium (kalium), Cu for copper (cuprum), Ag for silver (argentum), Au for gold (aurum), Hg for mercury (hydrargyrum), Pb for lead (plumbum), Sn for tin (stannum), Sb for antimony (stibium), W for tungsten (wolfram).

Care with capitals matters: Co is cobalt, but CO is carbon monoxide, a compound of carbon and oxygen. A symbol stands for three things: the name of the element, one atom of it, and, as the next topic shows, a definite mass of it — the atomic mass. Thus H means hydrogen, one atom of hydrogen, and 1 unit of mass of hydrogen. The student is expected to know the symbols of the first twenty elements and of the common metals; a table of these is given in the examples.

Dalton's most useful idea was that each element has a characteristic atomic mass. But an atom is so small — a hydrogen atom has a mass of about 1.67 × 10⁻²⁴ g — that it is inconvenient to use grams. Chemists therefore measure atomic masses relative to a chosen standard atom.

Dalton chose hydrogen, the lightest atom, as the unit, and then oxygen, 16 times heavier, was used for a long time because it combines with most elements and made determinations convenient. In 1961 the standard was changed by universal agreement to the carbon-12 atom. One atom of carbon-12 is assigned a mass of exactly 12 units, and one-twelfth of its mass is defined as one atomic mass unit, now written u (earlier amu). The relative atomic mass of an element is then the average mass of its atoms compared with one-twelfth of the mass of a carbon-12 atom.

Atomic mass therefore has no unit in the strict sense — it is a ratio — though the unit u is written to remind us of the standard. The atomic masses of the common elements, which the student must memorise for calculations, are: hydrogen 1, helium 4, carbon 12, nitrogen 14, oxygen 16, sodium 23, magnesium 24, aluminium 27, phosphorus 31, sulphur 32, chlorine 35.5, potassium 39, calcium 40, iron 56, copper 63.5, zinc 65, silver 108, iodine 127. The odd value for chlorine, 35.5, is the average of the two isotopes chlorine-35 and chlorine-37 in the proportion in which they occur; the next chapter explains isotopes.

An analogy helps. Imagine a fruit seller who has no weights but many identical water melons; he can still say a pumpkin weighs three melons and a papaya half a melon, and if he later learns that a melon is 2 kg he can convert everything to kilograms. The melon is the standard, as carbon-12 is for atoms. And in the same way, since the mass of one atom of carbon-12 is known in grams (1.993 × 10⁻²³ g), one u is 1.66 × 10⁻²⁴ g, and any atomic mass can be converted to grams if needed — but chemists rarely do so, because they work with vast numbers of atoms using the mole, introduced at the end of the chapter.

The atomic mass unit connects the two statements of Dalton's theory that puzzled the previous topics: since each atom has a fixed relative mass, the fixed number ratio of atoms in a compound gives a fixed mass ratio — the law of constant proportions in numbers. Water is H₂O: 2 × 1 = 2 units of hydrogen to 16 units of oxygen, the ratio 1 : 8.

Atoms of most elements are not able to exist alone; they are found combined with other atoms — of the same element or of different elements — as molecules or as ions. The exception is the noble gases: helium, neon, argon, krypton, xenon and radon exist as single atoms.

A molecule is a group of two or more atoms chemically bonded together, and it is the smallest particle of an element or compound that can exist independently and show all the properties of that substance. The atoms in a molecule are held together by attractive forces called chemical bonds, whose nature is studied in the next chapter.

Molecules of elements. The atoms of one element may join to form molecules of that element. The number of atoms in a molecule is called its atomicity. Monoatomic elements have a single atom as their molecule: the noble gases helium (He), neon (Ne), argon (Ar), and also metals such as sodium, iron and copper, whose atoms are held in a lattice and are conventionally represented by the symbol alone. Diatomic molecules have two atoms: hydrogen H₂, oxygen O₂, nitrogen N₂, chlorine Cl₂, fluorine F₂, bromine Br₂, iodine I₂. Triatomic: ozone O₃. Tetratomic: phosphorus P₄. Polyatomic: sulphur S₈. Carbon exists in giant structures — diamond and graphite — with no fixed molecule, so its symbol C is used.

Thus the gas oxygen is O₂, not O: a single oxygen atom does not exist free in air, and O₂ is the smallest particle of oxygen that does. This is why the formula of oxygen gas in every chemical equation is O₂, and why we say a molecule of hydrogen has two atoms.

A table for revision:

Molecules of compounds. Atoms of different elements join in definite proportions to form molecules of compounds: water H₂O has two hydrogen atoms and one oxygen atom (ratio 2 : 1 by number, 1 : 8 by mass); ammonia NH₃ has one nitrogen and three hydrogen atoms (14 : 3 by mass); carbon dioxide CO₂ has one carbon and two oxygen atoms (3 : 8 by mass); methane CH₄, hydrogen chloride HCl and sulphur dioxide SO₂ are other examples. A molecule of a compound is the smallest particle of the compound that shows its properties: one molecule of water is the least amount of water that is still water; split it and you have hydrogen and oxygen, not water. The molecular formula, which counts the atoms of each kind in one molecule, is the subject of the topics that follow on valency and formula writing.

Not every compound is made of molecules. Common salt, sodium chloride, is not made of NaCl molecules floating separately; it is a giant regular arrangement of charged particles. Compounds formed between a metal and a non-metal generally consist of such charged particles, called ions.

An ion is an atom or a group of atoms carrying an electric charge. An atom is normally electrically neutral; if it loses one or more electrons it becomes a positively charged ion, a cation; if it gains electrons it becomes a negatively charged ion, an anion. (The next chapter explains electrons; here it is enough to know that metals form cations and non-metals form anions.) In sodium chloride the sodium ion Na⁺ carries one unit of positive charge and the chloride ion Cl⁻ one unit of negative charge; the opposite charges attract, and the ions pack into a crystal in which each sodium ion is surrounded by chloride ions and each chloride by sodium ions. The formula NaCl tells the ratio of ions, 1 : 1, not the composition of a molecule. Such compounds are called ionic compounds.

A group of atoms carrying a charge is a polyatomic ion. The ammonium ion NH₄⁺ is a cation; the hydroxide ion OH⁻, the nitrate NO₃⁻, the carbonate CO₃²⁻, the sulphate SO₄²⁻ and the phosphate PO₄³⁻ are anions. The atoms in a polyatomic ion are held together tightly and the whole group behaves as a single unit in reactions and in formulae.

The charge on an ion is its valency — its combining capacity — and the tables below must be learnt, because formula writing depends on them.

Some metals show more than one valency; iron forms Fe²⁺ (ferrous, iron(II)) and Fe³⁺ (ferric, iron(III)), and copper forms Cu⁺ and Cu²⁺. The Roman numeral in the name gives the valency in that compound. Ions are the reason salt solutions conduct electricity and the reason ionic compounds have high melting points and are hard, brittle crystals — the strong attraction between opposite charges must be overcome. The chapter's next task is to combine these charged units into correct formulae.

The valency of an element is its combining capacity — the number of hydrogen atoms (or chlorine atoms) with which one atom of the element combines, or, for ions, the charge. Hydrogen has valency 1; oxygen, which combines with two hydrogens in water, has valency 2; nitrogen, three in ammonia, has 3; carbon, four in methane, has 4. Think of valency as the number of hands an atom has: an atom with two hands must hold two atoms with one hand each, or one atom with two hands. A chemical formula is the symbolic representation of the composition of a compound, and it is written so that the hands are all held — the total positive valency equals the total negative valency.

The rules for writing a formula:

1. Write the symbol of the metal or the positive ion (cation) first and the non-metal or negative ion (anion) second: NaCl, not ClNa. 2. Write the valency of each below or beside it. 3. Criss-cross the valencies: the valency of the first becomes the subscript of the second and vice versa. 4. If the subscripts have a common factor, divide by it to get the simplest ratio. 5. A subscript of 1 is not written. 6. If a polyatomic ion needs a subscript greater than 1, enclose it in brackets.

Worked examples.

Hydrogen chloride: H (1) and Cl (1); criss-cross gives H₁Cl₁ = HCl.

Magnesium chloride: Mg (2) and Cl (1); criss-cross: Mg₁Cl₂ = MgCl₂. Read it: one magnesium with two hands holds two chlorines with one hand each.

Aluminium oxide: Al (3) and O (2); criss-cross: Al₂O₃. Two aluminiums (6 hands) hold three oxygens (6 hands).

Calcium oxide: Ca (2) and O (2); criss-cross gives Ca₂O₂; divide by the common factor 2: CaO.

Sodium sulphide: Na (1), S (2): Na₂S. Carbon tetrachloride: C (4), Cl (1): CCl₄. Hydrogen sulphide: H (1), S (2): H₂S. Zinc oxide: Zn (2), O (2): ZnO. Iron(III) oxide: Fe (3), O (2): Fe₂O₃; iron(II) oxide: Fe (2), O (2): FeO.

For molecular (covalent) compounds of two non-metals the same method works when the valencies are taken from the number of hydrogens each combines with: water, H (1) and O (2), gives H₂O; ammonia, N (3) and H (1), gives NH₃; methane, C (4) and H (1), gives CH₄; carbon dioxide, C (4) and O (2), gives C₂O₄, simplified to CO₂.

The formula tells the number of atoms of each element in one molecule (or the ratio of ions in an ionic compound): CO₂ has one carbon and two oxygens; the subscript applies only to the symbol immediately before it. A number in front of a formula, a coefficient, multiplies the whole: 2H₂O is two molecules of water, four hydrogen atoms and two oxygen atoms in all. The next topic extends the method to polyatomic ions, where brackets are needed.

When one or both ions are polyatomic, the criss-cross method is the same, with one extra rule: a polyatomic ion that receives a subscript greater than 1 is enclosed in brackets, so that the subscript multiplies the whole group. A polyatomic ion is never broken up; it moves through formulae and reactions as one unit, like a single atom with several hands.

Worked examples.

Sodium nitrate: Na (1) and NO₃ (1); criss-cross gives Na₁(NO₃)₁ = NaNO₃. No brackets, since the subscript is 1.

Calcium hydroxide: Ca (2) and OH (1); criss-cross gives Ca₁(OH)₂ = Ca(OH)₂. The brackets show two hydroxide groups — two oxygens and two hydrogens. Written CaOH₂ it would wrongly mean one oxygen and two hydrogens.

Magnesium nitrate: Mg (2), NO₃ (1): Mg(NO₃)₂ — one magnesium, two nitrogens, six oxygens.

Sodium carbonate: Na (1), CO₃ (2): Na₂CO₃. No brackets, since the carbonate has subscript 1.

Ammonium sulphate: NH₄ (1), SO₄ (2): (NH₄)₂SO₄ — two ammonium groups, one sulphate.

Calcium phosphate: Ca (2), PO₄ (3): Ca₃(PO₄)₂ — three calciums (6 hands), two phosphates (6 hands).

Aluminium sulphate: Al (3), SO₄ (2): Al₂(SO₄)₃. Ammonium chloride: NH₄ (1), Cl (1): NH₄Cl. Calcium carbonate: Ca (2), CO₃ (2): Ca₂(CO₃)₂, reduced to CaCO₃. Potassium sulphate: K (1), SO₄ (2): K₂SO₄. Sodium hydrogen carbonate (baking soda): Na (1), HCO₃ (1): NaHCO₃. Copper(II) sulphate: Cu (2), SO₄ (2): CuSO₄.

Common acids follow the same pattern with hydrogen as the cation: hydrochloric acid HCl, nitric acid HNO₃, sulphuric acid H₂SO₄, carbonic acid H₂CO₃, phosphoric acid H₃PO₄. Common bases have the hydroxide anion: sodium hydroxide NaOH, potassium hydroxide KOH, calcium hydroxide Ca(OH)₂, aluminium hydroxide Al(OH)₃.

Reading a formula backwards. The examination also asks the reverse: given a formula, name the compound or count its atoms. Ca(OH)₂ contains 1 calcium, 2 oxygen and 2 hydrogen atoms, 5 in all. Al₂(SO₄)₃ contains 2 aluminium, 3 sulphur and 12 oxygen atoms, 17 in all. (NH₄)₂SO₄ contains 2 nitrogen, 8 hydrogen, 1 sulphur and 4 oxygen atoms, 15 in all. Counting correctly is the first step in the molecular mass calculations that follow.

The student should practise until the formulae of the twenty or so compounds in this topic can be written from memory of the valency tables alone; every later chapter of chemistry assumes this skill.

Once atomic masses and formulae are known, the mass of a molecule follows by addition. The molecular mass of a substance is the sum of the atomic masses of all the atoms in one molecule of the substance, expressed in atomic mass units (u). It is the relative mass of a molecule compared with one-twelfth of a carbon-12 atom.

Worked examples of molecular mass.

Water, H₂O: 2 hydrogen atoms (2 × 1 = 2) + 1 oxygen (16) = 18 u.

Carbon dioxide, CO₂: 12 + (2 × 16) = 12 + 32 = 44 u.

Ammonia, NH₃: 14 + (3 × 1) = 17 u.

Hydrogen chloride, HCl: 1 + 35.5 = 36.5 u.

Nitric acid, HNO₃: 1 + 14 + (3 × 16) = 63 u.

Sulphuric acid, H₂SO₄: (2 × 1) + 32 + (4 × 16) = 2 + 32 + 64 = 98 u.

Methane, CH₄: 12 + 4 = 16 u. Ethanol, C₂H₅OH: (2 × 12) + (6 × 1) + 16 = 46 u. Glucose, C₆H₁₂O₆: 72 + 12 + 96 = 180 u. Oxygen gas, O₂: 2 × 16 = 32 u; nitrogen N₂: 28 u; chlorine Cl₂: 71 u; hydrogen H₂: 2 u.

Ionic compounds do not consist of molecules, so the term molecular mass does not strictly apply. Instead we use the formula unit mass: the sum of the atomic masses of all the atoms in the formula unit of the compound, calculated exactly as a molecular mass is. The formula unit is the simplest ratio of ions given by the formula.

Worked examples of formula unit mass.

Sodium chloride, NaCl: 23 + 35.5 = 58.5 u.

Calcium chloride, CaCl₂: 40 + (2 × 35.5) = 111 u.

Calcium carbonate, CaCO₃: 40 + 12 + (3 × 16) = 100 u.

Sodium carbonate, Na₂CO₃: (2 × 23) + 12 + 48 = 106 u.

Zinc oxide, ZnO: 65 + 16 = 81 u. Potassium carbonate, K₂CO₃: 78 + 12 + 48 = 138 u. Calcium hydroxide, Ca(OH)₂: 40 + 2 × (16 + 1) = 74 u. Ammonium sulphate, (NH₄)₂SO₄: 2 × (14 + 4) + 32 + 64 = 132 u. Aluminium sulphate, Al₂(SO₄)₃: 54 + 3 × (32 + 64) = 342 u.

Two habits prevent the commonest mistakes. First, count every atom, including those inside brackets multiplied by the outside subscript. Second, write each product before adding: for H₂SO₄ write 2 × 1 = 2, 1 × 32 = 32, 4 × 16 = 64, total 98. The molecular mass is the bridge to the mole concept: the same number expressed in grams instead of atomic mass units is the mass of one mole of the substance.

Atoms and molecules are so small that any sample we can weigh contains a stupendous number of them. Chemists therefore count particles in a large unit, as a grocer counts eggs by the dozen or paper by the ream. The chemist's counting unit is the mole.

One mole of any species — atoms, molecules, ions or formula units — is that amount which contains as many particles as there are atoms in exactly 12 g of carbon-12. This number has been measured to be 6.022 × 10²³, and it is called the Avogadro number or Avogadro constant, N_A, in honour of the Italian scientist Amedeo Avogadro. A mole is thus a fixed number, 602,200,000,000,000,000,000,000, just as a dozen is 12; a mole of atoms is 6.022 × 10²³ atoms and a mole of molecules is 6.022 × 10²³ molecules. To picture the size of the number: a mole of rupee coins would cover the whole of India to a depth of several kilometres, and a mole of grains of rice would feed the world for centuries.

The great convenience of the mole is that it links number to mass. The mass of one mole of a substance, called its molar mass, is numerically equal to its atomic mass or molecular mass or formula unit mass, but expressed in grams instead of atomic mass units. Since the atomic mass of carbon is 12 u, one mole of carbon atoms has a mass of 12 g; since the atomic mass of oxygen is 16 u, one mole of oxygen atoms is 16 g and one mole of oxygen molecules (O₂, 32 u) is 32 g; one mole of water (18 u) is 18 g, one mole of sodium chloride (58.5 u) is 58.5 g. The atomic mass in grams is also called the gram atomic mass and the molecular mass in grams the gram molecular mass. Thus 18 g of water, 44 g of carbon dioxide and 32 g of oxygen each contain the same number of molecules, 6.022 × 10²³ — which is what makes the mole so useful in weighing out reactants for a reaction in the right proportions.

Three relations follow and form the basis of every numerical problem:

Number of moles = given mass ÷ molar mass, that is n = m / M.

Number of moles = given number of particles ÷ Avogadro number, that is n = N / N_A.

Number of particles = number of moles × 6.022 × 10²³.

So the mole is the meeting point of three quantities: a definite number of particles (6.022 × 10²³), a definite mass (the molar mass in grams), and the amount of substance. Any two can be converted through the third; the worked examples in the next topic show the routine. Historically the word mole comes from the Latin moles, a heap or pile, and the mole was adopted as the SI unit of amount of substance in 1971.

This topic drills the three conversions, since the examination sets at least one such problem every year. Use atomic masses H 1, C 12, N 14, O 16, Na 23, S 32, Cl 35.5, Ca 40, Fe 56, and N_A = 6.022 × 10²³.

Type 1: mass to moles. How many moles are in 100 g of water? Molar mass of H₂O = 18 g/mol. n = m / M = 100 / 18 = 5.56 mol.

How many moles are in 22 g of carbon dioxide? M = 44 g/mol; n = 22 / 44 = 0.5 mol.

Type 2: moles to mass. What is the mass of 0.5 mole of oxygen gas? M of O₂ = 32 g/mol; m = n × M = 0.5 × 32 = 16 g.

Mass of 3 moles of sodium chloride: m = 3 × 58.5 = 175.5 g. Mass of 0.2 mole of calcium carbonate: 0.2 × 100 = 20 g.

Type 3: moles to number of particles. How many molecules are in 2 moles of water? N = 2 × 6.022 × 10²³ = 1.204 × 10²⁴ molecules. How many atoms are in 0.5 mole of iron? 0.5 × 6.022 × 10²³ = 3.011 × 10²³ atoms.

Type 4: mass to number of particles (two steps: mass → moles → particles). How many molecules are in 9 g of water? n = 9 / 18 = 0.5 mol; N = 0.5 × 6.022 × 10²³ = 3.011 × 10²³ molecules. How many atoms are in 4 g of oxygen atoms? n = 4 / 16 = 0.25 mol; N = 0.25 × 6.022 × 10²³ = 1.505 × 10²³ atoms.

Type 5: number of particles to mass (particles → moles → mass). What is the mass of 3.011 × 10²³ molecules of carbon dioxide? n = 3.011 × 10²³ / 6.022 × 10²³ = 0.5 mol; m = 0.5 × 44 = 22 g. Mass of 12.044 × 10²³ atoms of sodium? n = 2 mol; m = 2 × 23 = 46 g.

Type 6: comparing. Which has more atoms, 100 g of sodium or 100 g of iron? Moles of Na = 100 / 23 = 4.35 mol; moles of Fe = 100 / 56 = 1.79 mol. Since the number of atoms is proportional to moles, 100 g of sodium has more atoms — about 2.6 × 10²⁴ against 1.1 × 10²⁴. This is the classic question: equal masses of lighter elements contain more atoms.

Type 7: atoms within molecules. How many atoms of hydrogen are in 0.5 mole of ammonia? Each NH₃ molecule has 3 hydrogen atoms, so 0.5 mol of NH₃ contains 1.5 mol of hydrogen atoms = 1.5 × 6.022 × 10²³ = 9.033 × 10²³ atoms. Total atoms in 1 mole of H₂SO₄: 7 atoms per molecule, so 7 × 6.022 × 10²³ = 4.215 × 10²⁴ atoms.

Method for full marks. Write the formula; compute the molar mass showing each atomic mass; write the relation used (n = m / M etc.); substitute; give the answer with its unit (mol, g, or a count). Powers of ten must be handled carefully: multiplying 0.5 by 6.022 × 10²³ gives 3.011 × 10²³, and 2 × 6.022 × 10²³ = 12.044 × 10²³ = 1.2044 × 10²⁴.

A formula carries quantitative information that can be turned into percentages, and the examination sometimes asks for the percentage composition of a compound — the percentage by mass of each element in it. This ties the chapter's formula writing, molecular mass and the law of constant proportions together in one calculation.

The method: find the molecular (or formula unit) mass; find the total mass contributed by each element (atomic mass × number of atoms); divide by the molecular mass and multiply by 100.

Worked example 1: water. H₂O has molecular mass 18 u. Mass of hydrogen = 2 × 1 = 2; percentage of hydrogen = (2 / 18) × 100 = 11.1 percent. Mass of oxygen = 16; percentage = (16 / 18) × 100 = 88.9 percent. The two add to 100 percent, and the ratio 11.1 : 88.9 is 1 : 8 — the law of constant proportions in a new dress.

Worked example 2: carbon dioxide. CO₂, 44 u. Carbon = 12; (12 / 44) × 100 = 27.3 percent. Oxygen = 32; (32 / 44) × 100 = 72.7 percent.

Worked example 3: ammonia. NH₃, 17 u. Nitrogen (14 / 17) × 100 = 82.4 percent; hydrogen (3 / 17) × 100 = 17.6 percent.

Worked example 4: calcium carbonate. CaCO₃, 100 u. Calcium 40 percent, carbon 12 percent, oxygen 48 percent — a convenient case because the formula unit mass is 100.

Worked example 5: urea, the fertiliser, CO(NH₂)₂ — molecular mass = 12 + 16 + 2 × (14 + 2) = 60 u. Nitrogen = 28; (28 / 60) × 100 = 46.7 percent. This is why urea bags are marked 46 percent N, and why urea is the most concentrated solid nitrogen fertiliser.

The reverse calculation is also useful: how much of an element is in a given mass of compound? Mass of oxygen in 90 g of water = 90 × (16 / 18) = 80 g; mass of calcium in 50 g of calcium carbonate = 50 × (40 / 100) = 20 g; mass of nitrogen in a 50 kg bag of urea = 50 × 0.467 = 23.3 kg.

Finally, a note on what a formula means. For a molecular compound the formula gives the actual number of atoms in one molecule (the molecular formula): glucose is C₆H₁₂O₆, not CH₂O, though the simplest ratio is 1 : 2 : 1. The simplest whole-number ratio is called the empirical formula, and for ionic compounds the formula unit is always the empirical formula — NaCl, CaCl₂, Al₂O₃ — because there are no molecules to count. Distinguishing the two is a matter for higher classes; at this level the student should be able to compute percentage composition and the mass of an element in a sample, and to recognise that a formula is a compact statement of the law of constant proportions.