What you'll learn
Atomic structure and the mole concept are the foundations of every calculation and explanation in CAPE Chemistry. An atom consists of a tiny, dense nucleus of protons and neutrons surrounded by electrons, and the mole is the chemist's counting unit that links the mass of a substance you can weigh to the number of particles it contains. This guide covers the subatomic particles and their properties, the behaviour of particles in electric fields, isotopes and relative atomic mass, the principles of the mass spectrometer, radioactivity and its uses, the mole, the Avogadro constant, empirical and molecular formulae, reacting masses, gas volumes, concentration and percentage yield. Worked examples are set in Caribbean contexts, from fertiliser production in Trinidad to bauxite processing in Jamaica.
In Unit 1 papers, mole calculations appear in almost every structured question, often hidden inside questions about other topics. Fluency here is the single biggest boost you can give your marks across the course.
Key terms and definitions
Proton — A positively charged particle in the nucleus, relative mass 1, relative charge +1.
Neutron — An uncharged particle in the nucleus, relative mass 1.
Electron — A negatively charged particle outside the nucleus, relative mass about 1/1840, relative charge −1.
Atomic (proton) number, Z — The number of protons in the nucleus of an atom.
Mass (nucleon) number, A — The total number of protons and neutrons in the nucleus.
Isotopes — Atoms of the same element with the same number of protons but different numbers of neutrons.
Relative atomic mass, Aᵣ — The weighted mean mass of an atom of an element compared with one-twelfth of the mass of an atom of carbon-12.
Relative molecular mass, Mᵣ — The weighted mean mass of a molecule compared with one-twelfth of the mass of a carbon-12 atom.
Mole — The amount of substance containing as many particles as there are atoms in exactly 12 g of carbon-12.
Avogadro constant, L or Nₐ — 6.02 × 10²³ particles per mole.
Molar mass — The mass of one mole of a substance, in g mol⁻¹.
Empirical formula — The simplest whole-number ratio of atoms of each element in a compound.
Molecular formula — The actual number of atoms of each element in one molecule.
Molar gas volume — The volume occupied by one mole of any gas at a stated temperature and pressure (22.4 dm³ at s.t.p., 0 °C and 1 atm; about 24 dm³ at room temperature and pressure).
Core concepts
The subatomic particles
| Particle | Relative mass | Relative charge | Location |
|---|---|---|---|
| Proton | 1 | +1 | Nucleus |
| Neutron | 1 | 0 | Nucleus |
| Electron | 1/1840 | −1 | Orbitals around the nucleus |
Almost all the mass of an atom is in the nucleus, but the nucleus occupies only a tiny fraction of its volume; most of an atom is empty space. Evidence for this came from Rutherford's gold foil experiment, in which most alpha particles passed straight through a thin sheet of gold while a few were deflected through large angles by the small, dense, positive nucleus.
Behaviour in electric fields
When beams of particles pass between charged plates:
- protons are deflected towards the negative plate;
- electrons are deflected towards the positive plate, and much more strongly than protons because their mass is far smaller;
- neutrons are not deflected because they are uncharged.
The angle of deflection is proportional to charge ÷ mass, which is why electrons bend so sharply.
Isotopes
Isotopes of an element have the same number of protons and electrons, so they have the same chemical properties, but different numbers of neutrons, so they have different masses and slightly different physical properties such as density and rate of diffusion. Chlorine, for example, exists mainly as chlorine-35 and chlorine-37.
Relative atomic mass from isotopic abundance
Aᵣ is a weighted average:
Aᵣ = Σ (isotopic mass × percentage abundance) ÷ 100
For chlorine, with about 75% chlorine-35 and 25% chlorine-37:
Aᵣ = (35 × 75 + 37 × 25) ÷ 100 = (2625 + 925) ÷ 100 = 35.5
The mass spectrometer
A mass spectrometer measures the masses and relative abundances of isotopes. The main stages are:
- Vaporisation — the sample is turned into a gas.
- Ionisation — high-energy electrons knock electrons off the atoms, forming positive ions.
- Acceleration — the ions are accelerated by an electric field.
- Deflection — a magnetic field deflects the ions according to their mass-to-charge ratio (m/z); lighter ions are deflected more.
- Detection — ions strike a detector, producing a spectrum of m/z against relative abundance.
The spectrum allows Aᵣ to be calculated, and for molecules, the molecular ion peak gives the Mᵣ.
Radioactivity
Some nuclei are unstable and decay, emitting:
- alpha (α) particles — helium nuclei (2 protons, 2 neutrons); stopped by paper; strongly ionising;
- beta (β) particles — fast electrons from the nucleus; stopped by a few millimetres of aluminium;
- gamma (γ) rays — high-energy electromagnetic radiation; reduced by thick lead or concrete.
Uses include carbon-14 dating of archaeological remains, cobalt-60 in cancer treatment and sterilising medical equipment, iodine-131 in thyroid diagnosis, radioactive tracers to detect leaks in pipelines, and smoke detectors containing americium-241.
The mole and the Avogadro constant
The mole lets chemists count particles by weighing. One mole of any substance contains 6.02 × 10²³ particles. The key relationships are:
- amount (mol) = mass (g) ÷ molar mass (g mol⁻¹)
- number of particles = amount × 6.02 × 10²³
- amount of gas (mol) = volume (dm³) ÷ molar gas volume
- amount in solution (mol) = concentration (mol dm⁻³) × volume (dm³)
Remember to convert cm³ to dm³ by dividing by 1000.
Empirical and molecular formulae
To find an empirical formula from percentage composition:
- Assume 100 g, so percentages become masses.
- Divide each mass by the element's Aᵣ to find moles.
- Divide by the smallest number of moles to find the ratio.
- Multiply to obtain whole numbers if necessary.
The molecular formula is a whole-number multiple of the empirical formula, found using Mᵣ: multiple = Mᵣ ÷ (empirical formula mass).
Reacting masses and limiting reagents
Balanced equations give mole ratios. To calculate a reacting mass: convert the known mass to moles, use the mole ratio, then convert to the required quantity. When two reactants are given, find which is limiting — the one that runs out first — and base the calculation on it.
Percentage yield and atom economy
- Percentage yield = (actual yield ÷ theoretical yield) × 100
- Atom economy = (Mᵣ of desired product ÷ total Mᵣ of all products) × 100
High atom economy is a goal of green chemistry, because fewer atoms are wasted as by-products.
Worked examples
Example 1: Relative atomic mass
Boron consists of 19.9% boron-10 and 80.1% boron-11. Calculate its relative atomic mass.
Aᵣ = (10 × 19.9 + 11 × 80.1) ÷ 100 = (199 + 881.1) ÷ 100 = 10.8 (to 3 significant figures).
Example 2: Empirical and molecular formula
A compound contains 40.0% carbon, 6.7% hydrogen and 53.3% oxygen by mass. Its Mᵣ is 180. Find its molecular formula. (Aᵣ: C = 12, H = 1, O = 16)
| C | H | O | |
|---|---|---|---|
| Mass (g) | 40.0 | 6.7 | 53.3 |
| Moles | 40.0 ÷ 12 = 3.33 | 6.7 ÷ 1 = 6.7 | 53.3 ÷ 16 = 3.33 |
| Ratio | 1 | 2 | 1 |
Empirical formula = CH₂O, with mass 12 + 2 + 16 = 30. Multiple = 180 ÷ 30 = 6. Molecular formula = C₆H₁₂O₆ (glucose — the sugar fermented in Caribbean rum distilleries).
Example 3: Reacting masses in fertiliser production
Ammonia from Point Lisas, Trinidad, is converted to ammonium sulfate fertiliser: 2NH₃ + H₂SO₄ → (NH₄)₂SO₄. What mass of ammonium sulfate can be made from 34.0 kg of ammonia? (Mᵣ: NH₃ = 17, (NH₄)₂SO₄ = 132)
Moles of NH₃ = 34 000 g ÷ 17 = 2000 mol. Ratio 2 NH₃ : 1 (NH₄)₂SO₄, so moles of product = 1000 mol. Mass = 1000 × 132 = 132 000 g = 132 kg.
Example 4: Concentration and titration
25.0 cm³ of sodium hydroxide is neutralised by 20.0 cm³ of 0.100 mol dm⁻³ hydrochloric acid. Find the concentration of the NaOH.
Moles HCl = 0.100 × 20.0 ÷ 1000 = 0.00200 mol. NaOH + HCl → NaCl + H₂O (1:1), so moles NaOH = 0.00200 mol. Concentration = 0.00200 ÷ (25.0 ÷ 1000) = 0.0800 mol dm⁻³.
Example 5: Percentage yield
In a laboratory preparation, the theoretical yield of aspirin is 3.60 g, but 2.70 g is obtained. Calculate the percentage yield.
Percentage yield = 2.70 ÷ 3.60 × 100 = 75.0%.
Example 6: Gas volumes
What volume of hydrogen is produced at room temperature and pressure when 6.0 g of magnesium reacts with excess hydrochloric acid? (Aᵣ Mg = 24; molar gas volume = 24 dm³ at r.t.p.)
Mg + 2HCl → MgCl₂ + H₂. Moles Mg = 6.0 ÷ 24 = 0.25 mol = moles H₂. Volume = 0.25 × 24 = 6.0 dm³.
Example 7: Radioactive half-life
Carbon-14 has a half-life of 5730 years. A sample originally containing 80 mg of carbon-14 is left for 17 190 years. What mass of carbon-14 remains?
17 190 ÷ 5730 = 3 half-lives. Mass remaining = 80 × (½)³ = 10 mg. Measuring the remaining carbon-14 in once-living material such as wood or bone allows its age to be estimated.
Example 8: Limiting reagent
5.6 g of iron is heated with 4.0 g of sulfur to form iron(II) sulfide. Which reactant is limiting, and what mass of FeS forms? (Aᵣ Fe = 56, S = 32)
Fe + S → FeS. Moles Fe = 5.6 ÷ 56 = 0.10 mol; moles S = 4.0 ÷ 32 = 0.125 mol. The ratio is 1 : 1, so iron is limiting (0.10 mol) and sulfur is in excess. Mass of FeS = 0.10 × (56 + 32) = 8.8 g.
Key formulae at a glance
| Quantity | Formula |
|---|---|
| Amount from mass | n = m ÷ M |
| Number of particles | N = n × 6.02 × 10²³ |
| Amount of gas | n = V ÷ 24 (r.t.p.) or V ÷ 22.4 (s.t.p.), in dm³ |
| Amount in solution | n = c × V (V in dm³) |
| Relative atomic mass | Aᵣ = Σ(isotopic mass × % abundance) ÷ 100 |
| Percentage yield | (actual ÷ theoretical) × 100 |
| Atom economy | (Mᵣ desired product ÷ Mᵣ of all products) × 100 |
Keeping this table at the front of your revision notes will speed up every calculation in the course.
Common mistakes and how to avoid them
- Forgetting to convert cm³ to dm³ in concentration and gas calculations.
- Using the wrong mole ratio — always balance the equation first.
- Rounding too early — keep extra significant figures until the final answer.
- Confusing mass number with relative atomic mass — A is a whole number for one isotope; Aᵣ is a weighted average.
- Saying isotopes have different chemical properties — they do not, because their electron arrangements are identical.
- Omitting units — every answer needs units such as mol, g, dm³ or mol dm⁻³.
How this links to your Internal Assessment
Mole calculations underpin most CAPE Chemistry practical work. In your School-Based Assessment you are likely to carry out titrations, determine the formula of a hydrated salt by heating, or measure gas volumes produced in a reaction. Examiners reward clear tables of results with correct units and precision, calculations set out step by step, and evaluation of errors — for example, explaining how incomplete heating would make the calculated number of water molecules too low. Practise processing your own experimental data with the same methods shown above.
Exam technique for Atomic structure and the mole concept
- Show every step of a calculation; method marks are awarded even if the final answer is wrong.
- Give answers to the appropriate number of significant figures, usually matching the data.
- For describe questions on the mass spectrometer, give the stages in order with what happens at each.
- For explain questions on deflection, refer to charge-to-mass ratio.
- Read the question carefully for whether gas volumes are at s.t.p. or room conditions.
Quick revision summary
- Protons (+1, mass 1) and neutrons (0, mass 1) form the nucleus; electrons (−1, mass 1/1840) occupy orbitals.
- Deflection in an electric field depends on charge ÷ mass.
- Isotopes: same Z, different neutrons; same chemistry, different physical properties.
- Aᵣ = Σ(mass × abundance) ÷ 100; mass spectrometer: vaporise, ionise, accelerate, deflect, detect.
- α, β and γ radiation differ in nature, penetration and uses.
- n = m ÷ M; particles = n × 6.02 × 10²³; n = V ÷ molar volume; n = c × V.
- Empirical formula from percentage composition; molecular formula from Mᵣ.
- Percentage yield and atom economy measure efficiency.