Why chemists need the mole

Atoms, ions and molecules are much too small to count one at a time in the laboratory. Reactions, though, still happen particle by particle. The mole gives chemists the link between that invisible particle scale and the masses and volumes we can actually measure.

A mole is an SI unit of amount of substance that contains exactly $6.02214076 \times 10^{23}$ specified elementary entities. The unit symbol is mol. An amount of substance is a physical quantity that measures how many specified elementary entities are present, using the mole as its unit.

An elementary entity is the particle, or specified group of particles, being counted. It may be an atom, molecule, ion, electron, formula unit, or a stated group such as pairs of ions. Be clear about the entity named: 1 mol of oxygen atoms is not the same thing as 1 mol of oxygen molecules.

Different substances all contain the same number of specified entities in 1 mol samples, but their masses are not the same. That is why 1 mol of aluminium, 1 mol of water molecules and 1 mol of sodium chloride formula units appear as very different quantities on the bench.

Avogadro constant as a conversion factor

The Avogadro constant is the physical constant equal to the number of elementary entities per mole. Its symbol is $N_a$ and its unit is $mol^{-1}$. The data booklet gives its value.

The particle–mole relationship is:

To go from moles to particles, multiply by $N_a$. To go from particles to moles, divide by $N_a$. It works like converting between kilograms and grams; the conversion factor is just enormous.

Specified entities matter

If 0.250 mol of $H_2O$ molecules is present, there are 0.250 mol of water molecules. Each water molecule contains three atoms, so the same sample contains 0.500 mol of hydrogen atoms and 0.250 mol of oxygen atoms, or 0.750 mol of atoms in total. Read the wording carefully: atoms, molecules, ions and electrons are different things to count.

The carbon-12 relative mass scale

Atomic masses are extremely small, so chemists use a relative scale instead. Relative atomic mass is the weighted mean mass of the atoms of an element compared with one-twelfth of the mass of an atom of carbon-12. The symbol is $A_r$. It has no unit, since it’s a ratio.

Relative formula mass means the mass of one formula unit of a substance compared with one-twelfth of the mass of an atom of carbon-12. Its symbol is $M_r$, and it has no unit either. Formula is the useful word here because it works for molecular substances, ionic compounds and giant covalent substances.

A formula unit is the simplest whole-number collection of particles shown by the formula of a substance. In an ionic compound such as $MgCl_2$, the formula unit has one $Mg^{2+}$ ion and two $Cl^-$ ions, although the solid itself is a giant lattice, not a set of separate molecules.

Calculating relative formula mass

In calculations, use the relative atomic masses from the data booklet to two decimal places, as required by the guide.

For example, for $Ca(NO_3)_2$:

The brackets matter. The subscript $2$ outside $(NO_3)_2$ doubles everything inside the bracket. Hydrates follow the same rule: in $CuSO_4 \cdot 5H_2O$, the $5$ applies to the whole $H_2O$ group.

Relative masses are ratios, so don’t write g or g mol$^{-1}$ after $A_r$ or $M_r$. Keep units for molar mass in the next section.

Atomic mass trends and properties

Going down a group in the periodic table, atoms generally have larger relative atomic masses because they contain more protons and neutrons. This increase can help explain some properties, including density trends and the mass contribution to melting or boiling behaviour. Don’t give mass too much credit, though: reactivity, metallic character and ion formation are usually explained more directly by electronic structure, nuclear charge and atomic radius.

Molar mass

Molar mass is the mass of one mole of a substance. Its symbol is M, and in this topic the usual chemistry unit is $\mathrm{g\,mol^{-1}}$. For IB calculations, its numerical value matches $A_r$ for atoms and $M_r$ for formulas, to the precision needed.

For example, if $M_r(H_2O) = 18.02$, then $M(H_2O) = 18.02\ \mathrm{g\,mol^{-1}}$. The first number is a unitless relative ratio. The second is a mass per mole.

The mass–mole relationship

Use this key relationship:

Rearrange it when the question needs a different quantity:

• $m = nM$ when you need a mass from an amount.
• $M = \frac{m}{n}$ when you need a molar mass from experimental mass and amount data.

Watch the units. If $M$ is in $\mathrm{g\,mol^{-1}}$, then $m$ must be in $\mathrm{g}$. If the question gives kg or mg, convert before using the equation.

Linking mass to number of particles

Most calculations in this topic join two conversions: mass $\leftrightarrow$ amount in mol $\leftrightarrow$ number of particles

Use $n = \frac{m}{M}$ to go from mass to moles, then $N = nN_a$ to go from moles to particles. For compounds, one formula can contain several atoms of a particular element. One mole of $Al_2O_3$ formula units contains 2 mol of aluminium ions and 3 mol of oxide ions.

Why this matters for chemical equations

A balanced chemical equation gives mole ratios, not mass ratios. Molar mass lets you take a measured mass of reactant, convert it into moles, use the coefficients in the equation, and then convert the product amount back into mass. That pathway is used to predict masses of products in stoichiometry.

Empirical formula and molecular formula

An empirical formula gives the simplest whole-number ratio of atoms of each element in a compound. A molecular formula gives the actual number of atoms of each element in one molecule.

For molecular substances, these two formulas may not match. $C_6H_{12}O_6$ has empirical formula $CH_2O$, since $6:12:6$ simplifies to $1:2:1$. With ionic compounds, the formula normally used is already an empirical formula because it shows the simplest ratio of ions in the lattice. Examples comparing molecular formulas with empirical formulas.

From percentage composition to empirical formula

Percentage composition by mass is the percentage of a compound’s mass that comes from each element. For an element X in a compound:

To find an empirical formula from percentages, follow this routine:

1. Assume 100 g of compound, so the percentages become masses in grams.
2. Convert each element’s mass into moles using $n = \frac{m}{M}$.
3. Divide all mole values by the smallest mole value.
4. If needed, multiply the ratio to reach whole numbers.
5. Write the empirical formula using the simplest whole-number ratio.

Approximation matters here, but it has to be sensible. Ratios like 1.00, 1.99 and 3.01 can be rounded to 1, 2 and 3 because small deviations are expected from measurement uncertainty. A ratio such as 1.33 should not be rounded to 1; it usually points to multiplying all terms by 3 to give a whole-number ratio.

From empirical formula to molecular formula

The molecular formula is a whole-number multiple of the empirical formula. Find the multiplier using:

Then multiply every subscript in the empirical formula by $k$. If the data are reliable, $k$ should be very close to a whole number.

Experimental mass changes and empirical formulas

Mass changes in reactions can be used to work out empirical formulas. In a simple oxide experiment, a known mass of metal is heated in air until the mass becomes constant. The increase in mass is the mass of oxygen that combined with the metal. Convert the metal mass and oxygen mass into moles, then find the simplest ratio.

The repeated heat–cool–weigh cycle is there for a reason. Heating to constant mass gives evidence that the reaction is complete; without it, the calculated formula may contain too little oxygen. A lid reduces loss of solid while still allowing oxygen to enter. Realistic improvements include using a more precise balance, heating for longer, controlling air access better, and avoiding loss of powder when lifting the lid.

Combustion analysis uses the same mole logic. In complete combustion of a compound containing carbon and hydrogen, all carbon atoms end up in $CO_2$ and all hydrogen atoms end up in $H_2O$. So the measured masses of $CO_2$ and $H_2O$ allow the moles of C and H in the original compound to be calculated.

A graph can show fixed composition too. If different groups heat different masses of magnesium, a plot of mass of magnesium oxide against mass of magnesium should be close to a straight line. Anomalies show experimental error, and the gradient reflects the constant mass ratio in the compound.

Solutions, solutes and solvents

A solution is a homogeneous mixture in which one or more solutes are spread evenly through a solvent. A solute is the substance that has dissolved in a solution. A solvent is the component that dissolves the solute and usually sets the physical state of the solution.

An aqueous solution has water as the solvent. In school chemistry, most concentration calculations use aqueous solutions because they are easy to prepare, transfer and mix accurately.

Words like concentrated and dilute are fine in everyday discussion, but they are too vague for calculations. A numerical concentration is much more useful: it tells another chemist exactly how much solute is present in a stated volume of solution.

Molar concentration

Molar concentration is the amount of solute per unit volume of solution. The guide uses this relationship:

You will also see the same relationship written as $C = n / V$. The volume must be the final volume of the solution, not just the volume of water added.

Square brackets show molar concentration. For example, $[Cl^-]$ means the molar concentration of chloride ions, and $[NaOH] = 0.200\ \text{mol dm}^{-3}$ means the concentration of sodium hydroxide is $0.200\ \text{mol dm}^{-3}$. The brackets refer to a particular solute or ion, not to the whole solution.

Units and mass concentration

Concentrations can also be written in $\text{g dm}^{-3}$. Mass concentration is the mass of solute per unit volume of solution.

Molar concentration and mass concentration are connected by molar mass:

$\rho = CM$

and therefore:

$C = \rho / M$

So grams per $\text{dm}^3$ can be converted into moles per $\text{dm}^3$ by using grams per mole.

Preparing standard solutions and choosing glassware

A standard solution has an accurately known concentration. To prepare one from a solid, weigh the solute, dissolve it in a small volume of deionized water, transfer it quantitatively into a volumetric flask, rinse the beaker and funnel into the flask, then make up to the calibration mark and mix thoroughly.

Glassware affects the uncertainty. Use a volumetric flask when you need an accurate fixed final volume. Use a volumetric pipette to transfer one accurate fixed volume. A measuring cylinder is faster but less precise, so it is not the best choice for preparing a high-quality standard solution.

A serial dilution is a sequence of dilutions where each new solution is made from the previous one. It helps when you need several lower concentrations from one stock solution, especially for calibration curves. The key idea is conservation of solute: adding solvent changes volume and concentration, but not the amount of solute transferred.

For a dilution:

Calibration curves

A calibration curve is a graph linking a measured signal from known standard solutions to their concentrations. For a coloured solution, the signal is often absorbance from a colorimeter or spectrophotometer.

To use one, prepare standards that cover the likely concentration range, measure their absorbance, plot absorbance against concentration, and draw a best-fit line or curve. Then measure the absorbance of the unknown and read its concentration from the calibration curve. If the unknown is outside the reliable range, dilute it so it falls within the range, then account for the dilution.

Good calibration depends on sensible glassware and careful technique: clean cuvettes, a consistent wavelength, accurate standards, and concentrations that cover the unknown rather than clustering in one small region.

Avogadro’s law

Avogadro’s law says that equal volumes of gases, at the same temperature and pressure, contain equal numbers of molecules. Because the number of particles is proportional to the amount in moles, gas volume is proportional to amount in moles too, provided temperature and pressure stay fixed.

For two gases under the same conditions:

That’s why reacting gas volumes can be worked out directly from the coefficients in a balanced equation, as long as every gas volume is measured at the same temperature and pressure.

Using gas volume ratios

Start with the balanced equation. Its coefficients give the mole ratio. For gases under the same conditions, the same coefficients also give the volume ratio.

For example, if an equation shows 2A(g) reacting with 3B(g), then 2 volumes of A react with 3 volumes of B under the same conditions. So $20\ \mathrm{cm}^3$ of A would require $30\ \mathrm{cm}^3$ of B. You don’t need molar mass unless a mass is involved.

Watch the states. Avogadro’s law applies to gases, so you cannot use a gas volume ratio to find the volume of a liquid product or a solid reactant.

Link to ideal gases

Avogadro’s law is exact for ideal gases. A real gas has particles with finite volume and may experience intermolecular forces, which means it can deviate from ideal behaviour. Deviations are greatest at high pressure, where particle volume becomes significant, and low temperature, where attractions between particles matter more. Gases with stronger intermolecular forces or larger particles tend to deviate more under comparable conditions.