The mole and Avogadro constant

• The mole (mol) is the SI unit of amount of substance.

• 1 mol contains exactly $6.02 \times 10^{23}$ elementary entities (Avogadro constant, $N_A$).

• Elementary entities can be atoms, molecules, ions, electrons or a specified group of particles.

• Convert between amount in moles and number of particles using:

  • $N = nN_A$

  • $n = \dfrac{N}{N_A}$

• In exam questions, always check what particle is being counted.

This diagram shows the definition of one mole using carbon-12, linking 12 g of carbon-12 to Avogadro’s number of atoms. It is useful for visualizing why the mole connects mass to particle number. Source

Relative atomic mass, relative formula mass and molar mass

• Relative atomic mass, $A_r$ compares the average mass of an atom with $\frac{1}{12}$ of the mass of a carbon-12 atom.

• Relative formula mass, $M_r$ = sum of all the $A_r$ values in a formula.

• $A_r$ and $M_r$ have no units.

• Molar mass, $M$ is the mass of 1 mol of a substance and has units g mol$^{-1}$.

• Numerically, $M$ is usually the same as $M_r$, but $M$ has units and $M_r$ does not.

• Use data booklet $A_r$ values to 2 decimal places in calculations.

Core mole calculations

• Key relationship: $n = \dfrac{m}{M}$

• Rearrangements:

  • $m = nM$

  • $M = \dfrac{m}{n}$

• Typical calculation route:

  • mass $\rightarrow$ moles $\rightarrow$ particles

  • particles $\rightarrow$ moles $\rightarrow$ mass

• Common exam trap: do not confuse number of moles with number of particles.

• Always write units clearly: g, mol, g mol$^{-1}$.

Empirical formula and molecular formula

• Empirical formula = simplest whole-number ratio of atoms in a compound.

• Molecular formula = actual number of each type of atom in a molecule.

• To find an empirical formula from percentage composition:

  • Assume 100 g if percentages are given.

  • Convert each element’s mass to moles.

  • Divide all mole values by the smallest.

  • Convert to the simplest whole-number ratio.

• If ratios are not whole numbers, multiply all by the same factor:

  • 0.5 $\rightarrow \times 2$

  • 0.25 $\rightarrow \times 4$

  • 0.33 $\rightarrow \times 3$

  • 0.67 $\rightarrow \times 3$

• To find a molecular formula:

  • Find the empirical formula mass.

  • Calculate factor = $\dfrac{\text{molar mass}}{\text{empirical formula mass}}$.

  • Multiply all subscripts in the empirical formula by this whole-number factor.

Concentration of solutions

• Molar concentration depends on amount of solute and volume of solution.

• Main equation: $n = CV$

• Rearrangements:

  • $C = \dfrac{n}{V}$

  • $V = \dfrac{n}{C}$

• Square brackets show molar concentration, e.g. $[\text{NaOH}] = 0.100\ \text{mol dm}^{-3}$.

• Required concentration units:

  • mol dm$^{-3}$

  • g dm$^{-3}$

• Important conversion:

  • $\text{g dm}^{-3} = (\text{mol dm}^{-3}) \times M$

  • $\text{mol dm}^{-3} = \dfrac{\text{g dm}^{-3}}{M}$

• Remember: volume in concentration questions is usually required in dm$^3$, not cm$^3$.

• Conversion: $1000\ \text{cm}^3 = 1\ \text{dm}^3$.

The image shows a volumetric flask, the key apparatus for preparing a solution of known volume and concentration. It is directly relevant to concentration calculations and standard-solution practical work. Source

Gas volumes and Avogadro’s law

• Avogadro’s law: equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.

• For gases at the same $T$ and $P$, volume ratios = mole ratios.

• This allows mole calculations from gas volumes in reaction questions.

• Always compare gases only when temperature and pressure are the same.

• In stoichiometry, convert gas volumes to moles before using the balanced equation unless a direct volume ratio can be used.

This is helpful for seeing that gas volume is directly linked to amount of substance when temperature and pressure are fixed. It supports the IB idea that gas volume ratios can be used like mole ratios. Source

Problem-solving strategy for mole questions

• Start by identifying the data type given: mass, particles, concentration/volume, gas volume, or percentage composition.

• Convert the given quantity into moles first whenever possible.

• Use the balanced equation only after you have found moles of the known substance.

• Convert final moles into the required form: mass, particles, volume, or concentration.

• Check that the answer has the correct units and sensible significant figures.

Exam traps and quick reminders

• $A_r$ and $M_r$ have no units; molar mass does.

• Moles are not particles: use $N_A$ when converting.

• For solutions, convert cm$^3$ to dm$^3$ before using $n = CV$.

• In empirical formula questions, convert mass to moles, not directly to ratios.

• A molecular formula must be a whole-number multiple of the empirical formula.

• For gas-volume ratios, only compare gases under the same temperature and pressure.