Atoms and Molecules

Every substance you can see, touch, or breathe — water in a glass, the air you inhale, the iron in a nail — is built from particles too small to see even under a school microscope. Chemists call the basic building block an atom, and when atoms join together in fixed, tiny whole-number ratios, they form a molecule. This section asks the question that founded modern chemistry: how do we know matter is made of discrete particles at all, rather than being infinitely divisible, and once we accept that it is, how do we name the particles, count how many atoms sit in a molecule, and predict the exact formula of a compound before ever making it in a lab? The answer traces back to John Dalton in 1808, who reasoned from two laws chemists had already measured — mass is conserved in a reaction, and a compound always has the same fixed composition — to a bold claim: matter is made of indivisible atoms that combine in whole-number ratios. That single idea unlocked atomic symbols, molecular formulae, ions, valency, and the criss-cross method used to write formulae correctly every time.

The core insight

Dalton's atomic theory explains the laws of chemical combination by proposing that matter is made of indivisible atoms combining in fixed whole-number ratios, and this fixed-ratio idea becomes a mechanical formula-writing rule once each element or ion is assigned a valency (or charge) and the criss-cross method turns those numbers into subscripts.

The core idea

Dalton's atomic theory (1808) was built to explain two experimental laws already known to chemists: the Law of Conservation of Mass (mass before a reaction equals mass after) and the Law of Constant Proportions (a given compound always contains the same elements in the same fixed ratio by mass). Dalton proposed that all matter is made of tiny, indivisible particles called atoms; that atoms are neither created nor destroyed in a chemical reaction; that atoms of one element are identical in mass and properties while atoms of different elements differ; and that atoms combine in small whole-number ratios to form compounds whose composition is always constant. Postulate 2 (atoms are neither created nor destroyed) directly explains conservation of mass, while postulates 5 and 6 (fixed whole-number combining ratios) explain constant proportions — the theory was accepted precisely because it made these two independently discovered laws fall out as consequences of a single picture of matter. An atom is the smallest particle of an element that may or may not exist independently; atoms are extremely small, with radii on the order of nanometres,

1\ \text{nm} = 10^{-9}\ \text{m},

where: the metre is the SI base unit of length, so an atomic radius quoted as "0.1 nm" means one ten-billionth of a metre — far below what any optical microscope can resolve. Each element carries a one- or two-letter atomic symbol by IUPAC convention: the first letter is always capital and the second, if present, is always small, e.g. H (hydrogen), O (oxygen), Na (sodium, from Latin natrium), and Fe (iron, from ferrum). A molecule is the smallest particle of an element or compound capable of independent existence, showing all the properties of that substance. Molecules of elements are classified by atomicity, the number of atoms per molecule: monoatomic (He, Ar), diatomic (H $_2$ , O $_2$ , N $_2$ , Cl $_2$ ), triatomic (O $_3$ , ozone), and polyatomic (P $_4$ , S $_8$ ). Molecules of compounds combine atoms of different elements in a fixed ratio, e.g. H $_2$ O, CO $_2$ , NH $_3$ , CH $_4$ — the same fixed-ratio idea Dalton proposed for atoms now applies one level up, to how atoms group into molecules. An ion is an atom or a group of atoms carrying electric charge: a cation is positive (formed by losing electrons) and an anion is negative (formed by gaining electrons). Some ions are single atoms (Na $^+$ , Cl $^-$ ); others are polyatomic ions, a fixed group of atoms carrying one overall charge, such as $\text{NH}_4^+$ , $\text{OH}^-$ , $\text{SO}_4^{2-}$ , $\text{CO}_3^{2-}$ , $\text{NO}_3^-$ , and $\text{PO}_4^{3-}$ . At this level these are used from a memorised table for writing formulae, not derived from bonding theory.

Figure

A labelled diagram comparing a single oxygen atom that does not exist freely with an oxygen molecule of two bonded atoms that does exist independently

Why it works

The mechanical rule chemists use to write a correct compound formula, without needing to know any bonding theory, follows directly from Dalton's requirement that atoms combine in fixed small whole-number ratios once each element's combining power is known.

Step 1. Assign each element or ion a valency — its combining capacity — or, for an ion, use the magnitude of its charge. Valency and ionic charge magnitude play exactly the same numerical role when building a neutral formula, because a formula unit must have zero net charge overall.

Step 2. Apply the criss-cross method: the valency (or charge magnitude) of one element becomes the numerical subscript of the other, and vice versa. This works because to balance combining power (or charge) exactly, each atom needs as many partner atoms as its own valency demands, and each partner atom needs as many of the first as its valency demands — swapping the two numbers as subscripts is precisely what makes the total combining power balance on both sides.

Step 3. Simplify the resulting subscripts to the smallest whole-number ratio if they share a common factor, and enclose any polyatomic ion in brackets before writing a subscript greater than one on it, since the subscript must multiply the whole group, not just its last atom. In symbols, for element A with valency $a$ combining with element B of valency $b$ , the criss-cross gives

\text{A}_b\text{B}_a,

where: $a$ and $b$ are first swapped as shown and then divided by their highest common factor if one exists, giving the simplest whole-number formula — exactly the fixed ratio Dalton's theory demanded.

Worked example

Problem. Write the chemical formula of calcium hydroxide, given the calcium ion is $\text{Ca}^{2+}$ and the hydroxide ion is $\text{OH}^-$ .

Step 1. Write the ions with their charge magnitudes for criss-crossing: $\text{Ca}^{2+}$ has magnitude 2, and $\text{OH}^-$ has magnitude 1.

Step 2. Criss-cross the magnitudes as subscripts — the charge on Ca $^{2+}$ (2) becomes the subscript of the hydroxide group, and the charge on OH $^-$ (1) becomes the subscript of calcium, giving $\text{Ca}_1(\text{OH})_2$ .

Step 3. Drop the subscript 1 on calcium (it is never written), and keep the brackets around OH since its subscript is greater than one: $\text{Ca(OH)}_2$ .

Answer: the formula of calcium hydroxide is $\text{Ca(OH)}_2$ — brackets are required around the polyatomic hydroxide ion because two hydroxide groups are present in the formula unit.

Competitive stretch

Apply the same rule to ammonium sulfate, using $\text{NH}_4^+$ (charge magnitude 1) and $\text{SO}_4^{2-}$ (charge magnitude 2). Criss-crossing gives $(\text{NH}_4)_2(\text{SO}_4)_1$ ; dropping the subscript 1 on the sulfate group and keeping the necessary bracket around ammonium (since its subscript is 2) gives $(\text{NH}_4)_2\text{SO}_4$ — note that sulfate does not need brackets here because its own subscript is 1, but ammonium does because its subscript is 2.

Common traps

Confusing atom and molecule. An atom is the smallest particle of an element; a molecule is the smallest particle capable of independent existence — a single oxygen atom (O) does not normally exist freely, but a molecule of oxygen (O $_2$ ) does.
Forgetting brackets around a polyatomic ion. When a polyatomic ion's subscript exceeds 1, as in Ca(OH) $_2$ or (NH $_4$ ) $_2$ SO $_4$ , it must be enclosed in brackets before the subscript is added — writing "CaOH2" is wrong.
Not simplifying the criss-crossed ratio. If criss-crossing produces subscripts sharing a common factor (e.g. valency 4 and valency 2 giving X $_2$ Y $_4$ ), reduce them to the smallest whole-number ratio (XY $_2$ ).
Mixing up cation and anion. A cation loses electrons and becomes positive; an anion gains electrons and becomes negative — these are frequently swapped when a student is asked to classify a given ion.

Ask the tutor: Want to practice criss-crossing more valencies and ionic charges, or double-check when brackets around a polyatomic ion are really needed? Ask the AI tutor for step-by-step formula-writing drills.

Why does criss-crossing valencies actually give the correct formula, instead of it just being a memorised trick?

It works because a stable compound has its total combining power balanced on both sides. If element A has valency $a$ and B has valency $b$ , then to balance exactly, you need $b$ atoms of A for every $a$ atoms of B — which is exactly what swapping the numbers as subscripts produces. It is not an arbitrary trick; it is a shortcut for "each side's total valency must match," and simplifying the ratio afterward just removes any common factor so the formula reflects the smallest repeating unit.

If oxygen valency is 2 and hydrogen valency is 1, why is the formula for water H2O and not H1O2?

Criss-crossing hydrogen (valency 1) and oxygen (valency 2) puts oxygen's valency (2) on hydrogen and hydrogen's valency (1) on oxygen, giving H $_2$ O $_1$ , which simplifies to H $_2$ O since the subscript 1 is dropped. Writing H $_1$ O $_2$ would mean swapping the numbers onto the wrong atom — a common slip is criss-crossing backwards, so always double-check which element's valency lands on which symbol.

Are all polyatomic ions like NH4+ and SO4^2- built the same way as simple ions like Na+?

Numerically yes, for the purpose of writing formulae — you treat a polyatomic ion's overall charge exactly like a simple ion's charge and criss-cross it the same way. The difference is only that a polyatomic ion is itself a fixed group of atoms bonded together (not derived from bonding theory at this level, just memorised from a table), so if its subscript in the final formula exceeds one, the whole group must be enclosed in brackets to show the subscript multiplies every atom in the group.