Fundamental Concepts of Matter and the Concept of the Mole

Chemistry is the study of matter and the transformations it undergoes, and every idea in the subject — from a balanced equation to a pH curve — ultimately rests on being able to count atoms and molecules reliably. The trouble is immediate: a single drop of water contains on the order of $10^{21}$ molecules, far too many to count one by one, yet a chemist routinely needs to know exactly how many molecules are present in a sample sitting on a balance. This chapter builds the vocabulary and the single most important tool that resolves that difficulty — the mole — starting from the most basic classification of matter and ending with the arithmetic that links a mass measured in grams to a number of particles too large to visualise.

Two ideas anchor everything that follows. First, that matter is built from discrete, countable entities — atoms, molecules and ions — rather than being infinitely divisible. Second, that chemists have agreed on a relative mass scale, pegged to a single reference isotope, that lets every element's mass be compared on the same footing. Combining the two turns the impossible task of counting atoms into an ordinary weighing exercise. This chapter assumes no prior chemistry; the internal structure of the atom (protons, neutrons and electrons) is developed in full in a later chapter of this block, and the precise effect of isotopes on relative atomic mass is treated in the chapter on mass spectrometry. Likewise, this chapter stops short of using chemical formulas to find unknown compounds (the next chapter) or of balancing chemical equations (the chapter after that) — here the formula of a substance is always given, and the goal is simply to count.

1. Classifying Matter: Elements, Compounds and Mixtures

All matter observed under ordinary laboratory conditions can be sorted into a small number of categories, and doing so correctly is the first skill of the subject.

A pure substance has a fixed, uniform composition and a definite set of physical properties (melting point, density, and so on). Pure substances divide into two kinds:

• An element is a pure substance that cannot be broken down into anything chemically simpler; every atom in a sample of an element has the same number of protons. Oxygen gas, $\text{O}_2$, and solid copper, $\text{Cu}$, are both elements.
• A compound is a pure substance formed when atoms of two or more elements combine chemically in a fixed, definite ratio. Water, $\text{H}_2\text{O}$, and sodium chloride, $\text{NaCl}$, are compounds — and critically, a compound's properties are generally nothing like those of the elements that compose it: chlorine is a toxic gas and sodium a reactive metal, yet together they form table salt.

A mixture, by contrast, contains two or more pure substances that are not chemically combined and so retain their individual properties, in a composition that can vary from one sample to the next. A mixture is homogeneous if its composition is uniform throughout, down to the molecular scale (salt water, brass, air); it is heterogeneous if distinct regions with different compositions can be distinguished (sand stirred into water, a salad). Unlike compounds, mixtures can in principle always be separated back into their components by purely physical means — filtration, distillation, chromatography — because no chemical bonds need to be broken.

Key distinction. A compound is formed and broken only by a chemical change, and its component elements combine in one fixed ratio by mass. A mixture is formed and separated by a physical change, and its components can be present in any ratio.

2. Atoms, Molecules and Ions

The modern atomic picture of matter traces back to John Dalton's atomic theory (1803), whose original postulates were:

• All matter is composed of atoms, which are indivisible and indestructible.
• Atoms of a given element are all identical to one another in mass and properties; atoms of different elements differ in mass and properties.
• Atoms combine with one another in small, whole-number ratios to form compounds.

Each of the first two postulates was later revised by discoveries unavailable to Dalton himself: the discovery of subatomic particles (protons, neutrons and electrons, developed in a later chapter of this block) showed atoms are in fact divisible, and the discovery of isotopes (covered in a later chapter of this block) showed that atoms of a given element are not all identical in mass, only in proton number and chemical behaviour. The third postulate, on whole-number combining ratios, has held up unmodified and underlies the empirical formula work of the next chapter.

A further consequence of the atomic theory — that a chemical reaction only rearranges atoms, never creating, destroying, or changing the identity of any atom — is the law of conservation of mass: the total mass of the reactants must equal the total mass of the products. This is the physical principle that makes balancing a chemical equation possible at all — the subject of the next chapter but one.

The third postulate above, on whole-number combining ratios, is itself a generalisation of two experimental laws established just before Dalton's theory. The law of definite proportions states that a given compound always contains the same elements combined in the same fixed ratio by mass, however the sample was prepared: water is always $11.2\%$ hydrogen and $88.8\%$ oxygen by mass, never any other split. The law of multiple proportions extends this to pairs of compounds formed from the same two elements: when carbon and oxygen combine to form both $\text{CO}$ and $\text{CO}_2$, the different masses of oxygen that combine with a fixed mass of carbon stand in a simple ratio of small whole numbers (here, $1:2$) — exactly what is expected if both compounds are built from whole atoms in fixed ratios.

An atom is the smallest unit of an element that retains the chemical identity of that element. Every atom has an internal structure — a dense, positively charged nucleus surrounded by electrons — but the detailed physics of that structure (and the particles it is built from) is deferred to a later chapter; for the purposes of this chapter, an atom is simply the fundamental countable unit of an element.

A molecule is a discrete, electrically neutral group of two or more atoms held together by chemical bonds, such as $\text{O}_2$, $\text{CO}_2$ or $\text{H}_2\text{O}$. An ion is an atom or group of bonded atoms that carries an overall electric charge because it has gained or lost one or more electrons — a cation ($\text{Na}^+$) is positively charged, an anion ($\text{Cl}^-$) negatively charged. Many compounds, including ionic solids such as $\text{NaCl}$, contain no discrete molecules at all, but rather an extended lattice of ions; this distinction resurfaces in Section 4.

3. The Unified Atomic Mass Unit and Relative Atomic Mass

Atoms are far too light to weigh individually in grams — a single carbon atom has a mass of roughly $2\times 10^{-23}~\text{g}$ — so chemistry uses a relative mass scale instead of an absolute one. By international agreement, the reference point of this scale is a single, specific isotope: carbon-12. The unified atomic mass unit, symbol $\text{u}$, is defined as exactly one-twelfth of the mass of one atom of carbon-12.

The relative atomic mass, $A_r$, of an element is then the weighted average mass of its naturally occurring atoms, measured in these units, taking into account the relative abundance of each isotope the element possesses. Because $A_r$ is a ratio of two masses, it is a pure number with no unit of its own. The periodic table lists $A_r$ for every element — for example $A_r(\text{H}) = 1.01$, $A_r(\text{C}) = 12.01$, $A_r(\text{O}) = 16.00$, $A_r(\text{Na}) = 22.99$, $A_r(\text{Cl}) = 35.45$ — values measured with high precision by mass spectrometry, the technique and the isotope arithmetic behind it covered in full in a later chapter of this block.

Why $A_r(\text{C})$ is $12.01$, not exactly $12$. The reference isotope $^{12}\text{C}$ has a mass of exactly $12~\text{u}$ by definition, but naturally occurring carbon is a mixture of isotopes, overwhelmingly $^{12}\text{C}$ with a small admixture of heavier $^{13}\text{C}$. The tabulated $A_r(\text{C}) = 12.01$ is the abundance-weighted average of the whole natural mixture, which is why it is not a round number.

4. Relative Molecular Mass and Relative Formula Mass

For a molecular substance, the relative molecular mass, $M_r$, is the sum of the relative atomic masses of every atom in one molecule, exactly as its formula specifies. For an ionic compound, which has no discrete molecules, the analogous quantity is called the relative formula mass, built the same way from the formula unit — but in IB usage the symbol $M_r$ is conventionally applied to both, and this course follows that convention.

Worked example. Find the relative molecular mass of carbon dioxide, $\text{CO}_2$.

Reading the formula, one molecule of $\text{CO}_2$ contains one carbon atom and two oxygen atoms, so their relative atomic masses are added according to that ratio:

Worked example. Find the relative formula mass of calcium carbonate, $\text{CaCO}_3$.

5. The Mole: A Chemist's Counting Unit

Chemical reactions happen atom-to-atom and molecule-to-molecule, but laboratory measurements happen on a balance, in grams. The mole is the SI base unit built precisely to bridge the two: it is a fixed, enormous number of elementary entities, exactly analogous to how "a dozen" always means twelve items, whatever those items are.

This number is the Avogadro constant, $N_A$, named for Amedeo Avogadro, whose early-nineteenth-century hypothesis that equal volumes of gas at the same temperature and pressure contain equal numbers of molecules eventually led to it (the quantitative consequences of that hypothesis for gas volumes are developed in a later chapter of this block). Since the 2019 redefinition of the SI, $N_A$ is fixed by definition to the exact value above, so the mole is now defined purely by a count of entities, rather than — as in the older definition — by reference to the number of atoms in exactly $12~\text{g}$ of carbon-12.

Key distinction. "One mole" is meaningless without stating of what. One mole of $\text{O}$ atoms is not the same as one mole of $\text{O}_2$ molecules — the latter contains twice as many oxygen atoms. Always name the entity: atoms, molecules, ions or formula units.

6. Molar Mass

The molar mass, $M$, of a substance is the mass of exactly one mole of that substance, with SI unit $\text{g}\,\text{mol}^{-1}$. The reason the mole is such a convenient bridge is that, for every element or compound, the numerical value of its molar mass in $\text{g}\,\text{mol}^{-1}$ is the same number as its relative atomic or relative molecular mass — only the unit changes, from the dimensionless $\text{u}$-based ratio to an absolute mass in grams.

This equality is not a coincidence but a direct consequence of how the unified atomic mass unit and the Avogadro constant were originally chosen to fit together: one mole of carbon-12 atoms, each of mass exactly $12~\text{u}$, has a combined mass extremely close to exactly $12~\text{g}$. (After the 2019 redefinition, in which $N_A$ became an exactly fixed number rather than a measured one, this equality is no longer mathematically exact — it now holds to within about one part in a billion — a refinement far beyond anything this course requires, but a sign of how precisely the modern SI has been tied together.)

7. Mole Calculations from Mass: $n = \dfrac{m}{M}$

With molar mass established, the amount of substance, $n$ (measured in mol), present in a sample of known mass $m$ (in grams) follows immediately:

with the equivalent rearrangements $m = nM$ and $M = \dfrac{m}{n}$ available depending on which quantity is unknown.

Worked example. Calculate the amount, in mol, of calcium carbonate present in a $25.0~\text{g}$ sample of pure $\text{CaCO}_3$.

Given: $m = 25.0~\text{g}$; from Section 4, $M(\text{CaCO}_3) = 100.09~\text{g}\,\text{mol}^{-1}$.

The sample's mass was given to three significant figures, so the result is quoted to three as well: $0.250~\text{mol}$ of $\text{CaCO}_3$.

8. Relating Moles to Numbers of Particles: $N = nN_A$

Because the mole is defined as a fixed count, the number of individual entities $N$ in an amount $n$ follows by simple multiplication by the Avogadro constant:

Combining this with $n = \dfrac{m}{M}$ gives a single chain, mass $\to$ moles $\to$ number of particles, that recurs constantly throughout the course:

Worked example. How many water molecules are present in $3.60~\text{g}$ of water?

Step 1 — find the amount in moles, using $M(\text{H}_2\text{O}) = 2A_r(\text{H}) + A_r(\text{O}) = 2(1.01) + 16.00 = 18.02~\text{g}\,\text{mol}^{-1}$:

Step 2 — convert to a number of molecules:

Worked example — counting atoms within a formula. A sample contains $0.500~\text{mol}$ of $\text{CO}_2$. How many oxygen atoms does it contain?

Each $\text{CO}_2$ molecule, by its formula, contains two oxygen atoms for every one molecule, so the amount of oxygen atoms is twice the amount of $\text{CO}_2$ molecules:

Notice that the amount of carbon atoms in the same sample is simply $0.500~\text{mol}$, equal to the amount of $\text{CO}_2$ itself, since each molecule contributes only one carbon atom — the multiplying factor always comes directly from the subscript in the formula.

9. Common Exam Pitfalls

• Quoting a relative atomic or molecular mass with a unit. $A_r$ and $M_r$ are ratios of two masses and are therefore dimensionless; only molar mass $M$, measured in actual grams per mole, carries a unit.
• Forgetting to state which entity a mole count refers to. "Moles of oxygen" is ambiguous between $\text{O}$ atoms and $\text{O}_2$ molecules, and the two amounts of substance differ by a factor of two for the same mass of oxygen.
• Using the wrong number of atoms from a formula. In $\text{CaCO}_3$, every mole of compound supplies three moles of oxygen atoms, not one — the subscript multiplies through to the whole-formula amount.
• Mixing up $N$ and $n$. $n$ is an amount of substance, in mol; $N$ is a number of particles, a pure (and usually enormous) count. The Avogadro constant $N_A$, with its own unit $\text{mol}^{-1}$, is what converts one into the other.
• Treating a mixture as if it were a compound, or vice versa. A compound has one fixed composition by mass and requires a chemical change to separate; a mixture has variable composition and can always be separated physically.
• Carrying too few significant figures in $A_r$ values from the data booklet, then compounding the rounding error across a multi-step calculation. Use the full listed value of each $A_r$ and round only the final answer.

Summary

• Matter is classified into elements and compounds (both pure substances, with fixed composition) and mixtures (variable composition, physically separable); compounds alone require a chemical change to break apart.
• An atom is the fundamental countable unit of an element; a molecule is a neutral group of bonded atoms; an ion is a charged atom or group of atoms.
• The unified atomic mass unit, $1~\text{u} = \tfrac{1}{12}m(^{12}\text{C})$, is the reference against which relative atomic mass $A_r$ (and, built from it, relative molecular/formula mass $M_r$) is measured — both are dimensionless ratios.
• The mole is a fixed count of entities, $1~\text{mol} = N_A = 6.02214076\times 10^{23}$ entities, where $N_A$ is the Avogadro constant.
• Molar mass $M$ (unit $\text{g}\,\text{mol}^{-1}$) is numerically equal to $A_r$ or $M_r$, and links mass to amount of substance through $n = \dfrac{m}{M}$.
• The number of individual particles in a sample follows from $N = nN_A = \dfrac{m}{M}N_A$ — always state explicitly which entity (atom, molecule, ion or formula unit) is being counted.
• Using chemical formulas to find the composition of an unknown substance (empirical and molecular formulas) and using balanced equations to relate amounts of different substances in a reaction are developed in the next two chapters.