Look at a wall. It’s made of bricks, and each brick is made of baked clay. Now imagine you keep breaking it: the brick into chunks, the chunks into grains, the grains into smaller and smaller bits. Keep going and you always end up at the same place: atoms.
The dumb question of this lesson is: what is a wall made of? The answer is surprising: just three kinds of piece. They’re the same ones that make up your hand and the air you breathe. What changes is how many you use and how you put them together, just like a box of building blocks.
This lesson teaches four ideas: what’s inside an atom, what makes an atom one element or another, how the same pieces give very different things, and how you count atoms by the heap.
Pick your card: ♣ if you’re starting from scratch, ♦ if a simple formula doesn’t scare you, ♠ if you want the notation. You can switch at any time.
Inside the atom
Newbie
An atom has a nucleus in the middle, a tiny, heavy little ball, and a cloud of electrons around it.
The nucleus is made of two pieces:
- Protons, which have a positive electric charge (+).
- Neutrons, which have no charge.
Electrons have a negative charge (−) and are incredibly light: one proton weighs as much as about 1,800 electrons.
The strangest part is the size. If the nucleus were a blueberry, the whole atom would be a football stadium. Between the blueberry and the stands there’s nothing: an atom is almost all empty. And yet nearly all of its weight is in the blueberry.
Intermediate
An atom is on the order of 10⁻¹⁰ m across and its nucleus on the order of 10⁻¹⁵ m: the nucleus is about a hundred thousand times smaller in diameter. Masses are measured in atomic mass units (u), roughly what a proton weighs:
| Piece | Where it is | Charge | Mass |
|---|---|---|---|
| Proton | nucleus | +1 | 1.007 u |
| Neutron | nucleus | 0 | 1.009 u |
| Electron | around it | −1 | 0.000 55 u |
The protons and neutrons of the nucleus weigh almost the same, and a proton weighs as much as 1,836 electrons. That’s why more than 99.9% of an atom’s mass is in the nucleus.
How do we know there’s a nucleus if we can’t see it? In 1911, Rutherford, Geiger and Marsden fired α particles (tiny and positively charged) at a very thin sheet of gold. Almost all went straight through, but a few bounced back. That only makes sense if the positive charge is packed together, with almost all the mass in a tiny point, and the rest of the atom is almost all empty. The neutron was discovered later, in 1932.
Technical
| In ♦ we said | In ♠ we write |
|---|---|
| on the order of 10⁻¹⁵ m | R = r₀ A^(1/3), r₀ ≈ 1.2 fm (1 fm = 10⁻¹⁵ m) |
| the protons and neutrons of the nucleus | A: number of nucleons |
| atomic mass units (u) | u = m(¹²C)/12 = 1.660 539 068 92(52)×10⁻²⁷ kg |
| a proton weighs as much as 1,836 electrons | mₚ/mₑ = 1836.152 673 426(32) |
| positively charged | q = +e, e = 1.602 176 634×10⁻¹⁹ C (exact) |
| almost all empty | V_nucleus / V_atom ~ (10⁻¹⁵ / 10⁻¹⁰)³ = 10⁻¹⁵ |
| almost all the mass in a tiny point | nuclear density ρ ≈ 2.3×10¹⁷ kg/m³ |
Scattering measurements give a nuclear radius R = r₀ A^(1/3), with r₀ ≈ 1.2 fm and A the number of nucleons (OpenStax, University Physics vol. 3, 10.1). R growing as A^(1/3) means the volume grows as A: nucleons pack at constant density, like drops of a liquid that won’t compress.
That density comes from putting a mass u in a sphere of radius r₀: ρ ≈ u / (4/3 π r₀³) ≈ 2.3×10¹⁷ kg/m³, the same value OpenStax works out for iron-56 (HyperPhysics gives 2×10¹⁷). One cubic centimetre of nuclear matter would weigh about 2.3×10¹¹ kg.
With mₚ/mₑ = 1836.15 (CODATA 2022), in an atom with as many neutrons as protons the electrons are 0.03% of the mass. Volume goes the other way: with R_atom/R_nucleus ~ 10⁵, the nucleus fills ~10⁻¹⁵ of the volume.
Rutherford inferred the nucleus (Phil. Mag. 21, 669, 1911) from α particles scattered at large angles, impossible in an atom with its charge spread out. Chadwick identified the neutron (Nature 129, 312, 1932). Since 2019, e is exact by definition; u is, by definition, one twelfth of the mass of a ¹²C atom.
Elements and isotopes
Newbie
The number of protons is the atom’s ID card: it decides which element it is. One proton, hydrogen. Six, carbon. Eight, oxygen. Seventy-nine, gold. Change the number of protons and you change the element.
Neutrons are like weight in a backpack: they change how much the atom weighs, but not who it is. A carbon with 6, 7 or 8 neutrons is still carbon. Those versions are called isotopes, and they’re named after the total of protons plus neutrons: carbon-12, carbon-13, carbon-14.
Some isotopes last forever. Others don’t: they fall apart on their own, and that’s called radioactivity. Carbon-14 is like that: from any heap of carbon-14, half falls apart in about 5,700 years.
And the electrons? If an atom gains or loses some, it ends up with an electric charge (it’s an ion), but it’s still the same element.
Intermediate
Two numbers describe a nucleus:
- The atomic number (Z) is the number of protons. It identifies the element: Z = 6 is carbon, always.
- The mass number (A) is protons plus neutrons. The neutrons are A − Z.
It’s written with A on top and Z below, to the left of the symbol: ¹³₆C. Since the symbol already gives Z, it’s often left out: ¹³C, or “carbon-13”.
The isotopes of an element have the same Z and a different number of neutrons. They behave almost the same in chemistry; what changes is the mass and whether the nucleus holds. Natural carbon is a mixture: 98.9% ¹²C and 1.1% ¹³C, both stable (that’s their abundance). There are also traces of ¹⁴C, which forms high in the atmosphere and is radioactive: from any amount, half decays in about 5,700 years. That time is called the half-life.
An ion is an atom with more or fewer electrons than protons, and so it has a charge. Na⁺ has 11 protons and 10 electrons: charge +1, and it’s still sodium.
Technical
| In ♦ we said | In ♠ we write |
|---|---|
| atomic number (Z) | Z |
| the neutrons are A − Z | N = A − Z |
| mass number (A) | A = Z + N; nuclide ᴬ_Z X |
| whether the nucleus holds | chart of nuclides: stable and radioactive by (Z, N) |
| abundance | atom fraction xᵢ, with Σ xᵢ = 1 |
| half-life | t½: N(t) = N₀ · 2^(−t / t½) |
| charge | q = (Z − nₑ) · e |
A nuclide is fixed by (Z, N). The chart of nuclides lays them out with N on the horizontal axis and Z on the vertical. Between Z = 1 and 10, CIAAW lists 20 stable nuclides, all with N between Z − 1 and Z + 2: in light nuclei, stability hugs the diagonal N = Z. Beryllium and fluorine have just one each (⁹Be and ¹⁹F).
Being in the band isn’t enough: nuclides missing from that list, inside or outside it, decay. Each nucleus does so with a constant probability per unit time, so the number of nuclei follows N(t) = N₀ · 2^(−t/t½). Two examples, both inside the band:
- ³H (tritium): t½ = 12.32 years (OpenStax, table 10.1); TUNL recommends 12.323 ± 0.020 years. It turns into ³He.
- ¹⁴C: OpenStax gives 5,730 years and TUNL recommends 5,686 ± 40 (mean of recent measurements, 5,699 ± 20). With about 5,700 years, after 11,400 a quarter is left.
The abundances xᵢ aren’t universal constants: they depend on where the sample comes from. CIAAW gives ¹³C a range of 0.0096 to 0.0116 in normal terrestrial material.
An ion’s charge is q = (Z − nₑ)·e. Changing nₑ is chemistry (ionising); changing Z is nuclear physics.
Add neutrons to a carbon and it’s still carbon. To change the element you have to touch the protons, and that’s not chemistry anymore.
Try it: build an atom
Move the protons, neutrons and electrons. See which element you get, whether the nucleus holds and whether the atom ends up charged. Try 6 protons and 8 neutrons.
Same pieces, built differently
Newbie
With a box of identical blocks you can build a car or a castle. Atoms work the same way.
When several atoms hook together, they form a molecule. Water, for example, is one oxygen atom hooked to two hydrogen atoms. The hook is called a bond.
The best example is diamond and the lead of a pencil, which is called graphite. Both are just carbon. What changes is how the pieces are hooked together:
- In diamond, each atom hooks onto 4 neighbours, in every direction. Everything is locked tight, which is why it’s one of the hardest things there is.
- In graphite, each atom hooks onto 3 neighbours, all in the same plane. You get sheets stacked on top of each other that barely hold on to one another. When you write, the sheets slide off and stick to the paper.
So a thing doesn’t depend only on which pieces it has, but on how they’re put together.
Intermediate
A molecule is two or more atoms held together by chemical bonds. It’s written as a formula: H₂O is one oxygen and two hydrogens; O₂, two oxygens. A molecule can be made of a single element.
Atoms can also form huge networks instead of separate molecules. Pure carbon forms two well-known ones:
| Diamond | Graphite | |
|---|---|---|
| Neighbours of each atom | 4, at the corners of a tetrahedron | 3, in flat sheets of hexagons |
| Between layers | No layers: 3D network | Weak forces |
| Hardness | Among the hardest known | Soft: pencil lead |
| Electricity | Doesn’t conduct | Conducts |
In a graphite sheet, the centres of neighbouring atoms are 1.4×10⁻¹⁰ m apart.
Watch out for a trap: an element isn’t the same as a substance. Carbon is an element; diamond, graphite and CO₂ are different substances made from it. Changing the substance (rearranging the bonds) is chemistry; changing the element means touching the nucleus.
At 25 °C, graphite is the most stable form of carbon, and yet a diamond doesn’t turn into graphite: you’d have to break its bonds, and that takes a huge amount of energy. Why a change that “should” happen doesn’t is the subject of another lesson on this path.
Technical
| In ♦ we said | In ♠ we write |
|---|---|
| chemical bonds | single C–C bond (1.54 Å) and double C=C (1.34 Å) |
| 4, at the corners of a tetrahedron | sp³ hybridisation, angle arccos(−1/3) ≈ 109.5° |
| 3, in flat sheets | sp² hybridisation, 120° angle |
| 1.4×10⁻¹⁰ m | d = 1.4 Å (1 Å = 10⁻¹⁰ m) |
| weak forces | non-covalent interactions between layers |
| the most stable form | ΔG° < 0: ΔᵣG°(diamond → graphite) = −2.9 kJ/mol |
| break its bonds | C–C bond energy ≈ 345 kJ/mol |
| doesn't turn | metastable (kinetic stability) |
In diamond, each carbon is sp³ and bonds to 4 carbons at the corners of a tetrahedron; in graphite it’s sp² and bonds to 3 in the plane (UW–Madison, Chem 109). The tetrahedral angle comes from two vectors to opposite corners, (1, 1, 1) and (1, −1, −1): cos θ = −1/3, θ ≈ 109.47°. The three bonds of a hexagonal plane are at 120°.
OpenStax table 7.3 gives the single C–C bond an average length of 1.54 Å and 345 kJ/mol, and the double C=C bond 1.34 Å. The distance between neighbours in a graphite sheet, ~1.4 Å, falls between the two. The sheets are held only by non-covalent interactions, which is why graphite marks paper.
Thermodynamics at 25 °C: diamond’s Gibbs energy is 2.90 kJ/mol above graphite’s (OpenStax, appendix G), so ΔᵣG°(diamond → graphite) = −2.9 kJ/mol, the same figure UW–Madison gives. The sign says graphite is more stable. Kinetics says otherwise: rearranging the lattice means breaking C–C bonds of about 345 kJ/mol, some 120 times what is gained per mole. Diamond is metastable.
Counting atoms: the mole
Newbie
To count eggs we use dozens: 12. Atoms are so small that you need a far bigger pack: the mole, which is 6×10²³ things, a 6 with 23 zeros after it.
A mole of water molecules weighs about 18 grams: one sip. If you counted those molecules at one per second since the Big Bang, you wouldn’t have counted even a millionth of them.
A mole is always the same number of things, like a dozen. What changes is how much it weighs: a dozen eggs doesn’t weigh the same as a dozen watermelons.
Intermediate
The mole is the unit for counting particles. Since 2019, one mole is exactly 6.022 140 76×10²³ entities (atoms, molecules, ions…), Avogadro’s number. It used to be defined as the number of atoms in 12 g of carbon-12.
The handy part is that what a mole weighs, in grams per mole, has the same digits as the mass of one particle in u:
- Carbon: 12.011 g/mol.
- Water: 2 × 1.008 + 15.999 = 18.015 g/mol.
- Iron: 55.845 g/mol.
Example: a glass with 250 g of water is 250 / 18.015 ≈ 13.9 mol, that is, 13.9 × 6.022×10²³ ≈ 8.4×10²⁴ molecules.
Why does carbon weigh 12.011 and not exactly 12? Because natural carbon is a mixture of isotopes: almost all ¹²C, with a little ¹³C, which is heavier.
And a trap: a mole isn’t a mass, it’s a number of things. One mole of carbon is 12 g; one of iron, almost 56.
Technical
| In ♦ we said | In ♠ we write |
|---|---|
| exactly 6.022 140 76×10²³ | N_A = 6.022 140 76×10²³ mol⁻¹ |
| a number of things | amount of substance n = N / N_A |
| grams per mole | molar mass M = m / n |
| the same digits as the mass of one particle in u | M = Aᵣ · Mᵤ, Mᵤ = 1.000 000 001 05(31) g/mol |
| mixture of isotopes | Aᵣ(C) = Σᵢ xᵢ · Aᵣ(ⁱC) |
Amount of substance is n = N / N_A, with N_A fixed exactly since the 2019 SI redefinition. Molar mass is M = Aᵣ · Mᵤ. Before 2019, Mᵤ was 1 g/mol by definition. Now it’s measured: Mᵤ = 1.000 000 001 05(31) g/mol (CODATA 2022). The difference, one part in a billion, doesn’t show outside metrology. With that value, M(¹²C) = 12 Mᵤ ≈ 12.000 000 013 g/mol.
An element’s atomic weight is the mean of its isotopes’ masses, weighted by their abundance. For carbon, with NIST’s representative fractions, 0.9893 and 0.0107: Aᵣ(C) = 0.9893 · 12 + 0.0107 · 13.003 355 ≈ 12.0107. Since the abundance of ¹³C varies from sample to sample, CIAAW gives not a number but a range, [12.0096, 12.0116]. Routine calculations use the conventional value 12.011.
Final challenge
In three sentences
- Everything you touch is made of atoms, and every atom is three pieces: protons and neutrons in a tiny nucleus that holds almost all the mass, and electrons around it.
- Protons decide the element, neutrons the isotope and electrons the charge: changing neutrons or electrons doesn’t turn carbon into anything else.
- The same pieces built differently give different things (diamond and graphite are just carbon), and to count them we use the mole, a giant dozen of 6×10²³.
Sources
- Atom, nucleus, particles and sizes: OpenStax, Chemistry 2e, 2.2 and 2.3 (table 2.2, the blueberry and the stadium), cross-checked with OpenStax, University Physics vol. 1, 1.1. Molecules: OpenStax, 2.4.
- Nuclear radius and density, and half-lives: OpenStax, University Physics vol. 3, 10.1, cross-checked with HyperPhysics and with TUNL, ³H and ¹⁴C. Carbon-14 in the atmosphere: OpenStax, Chemistry 2e, 21.3.
- History: Rutherford, Phil. Mag. 21, 669 (1911), doi:10.1080/14786440508637080; Chadwick, Nature 129, 312 (1932), doi:10.1038/129312a0.
- Constants (CODATA 2022): mₚ/mₑ, u, Mᵤ and N_A. Definition of the mole and the elementary charge: BIPM, SI Brochure.
- Stable isotopes and abundances: CIAAW. Standard atomic weights (2024): CIAAW. Representative masses and fractions for carbon: NIST.
- Diamond and graphite: OpenStax, Chemistry 2e, 10.5 and UW–Madison, Chem 109, D26.1. Bonds: OpenStax, 7.5, table 7.3. Formation energies: OpenStax, appendix G.
- The mole as a dozen and molar mass: OpenStax, Chemistry 2e, 3.1.
How this lesson was made: researched and written with help from Claude, from the sources above, checked and cross-checked against a second source between 13 and 15 October 2026. Numbers without their own link are calculations from them. The drawings are our own.