3.1.2 Amount of Substance

When you see Aᵣ or Mᵣ, remember what the scale is doing: both compare mass with 1/12 of the mass of one carbon-12 atom. They are ratios, so they have no units.

This is a relative scale, so Aᵣ and Mᵣ are pure numbers. Molar mass uses the same numerical value but carries units, usually g mol⁻¹; keep those ideas separate in calculations.

If the substance is ionic, use relative formula mass for one formula unit because there are no discrete molecules. You calculate it in the usual way by adding the Aᵣ values in the formula.

For NaCl, for example, add Aᵣ(Na) + Aᵣ(Cl), but call the result relative formula mass because the solid is an ionic lattice of ions; it does not contain discrete NaCl molecules.

For the definition of Aᵣ, I’d learn this accurately: the weighted mean mass of an atom of an element compared with 1/12 of the mass of one carbon-12 atom. ‘Weighted mean’ matters because isotopes occur in different abundances.

If isotopes are present in unequal abundances, the arithmetic must reflect those abundances. A simple mean of the isotope mass numbers is only correct in the special case where their abundances are equal.

Relative molecular mass is the mean mass of a molecule compared with 1/12 of the mass of one carbon-12 atom. It is a ratio and therefore has no unit.

In a calculation, you obtain Mᵣ by adding the Aᵣ values for all atoms in the molecular formula. Keep that calculation rule separate from the definition: the definition is the carbon-12 mass comparison.

One mole contains the Avogadro constant number of specified particles. Always name the particle — atoms, molecules, ions, electrons or formula units — because one mole of a substance does not always mean one mole of atoms.

Nₐ links amount in moles to particle number: n moles contains nNₐ specified entities. You do not need to memorise the numerical value of Nₐ; use the value supplied when a calculation needs it.

Treat coefficients in a balanced equation as mole ratios. The same mole idea can count any specified entity, so read the wording carefully before deciding whether you need atoms, molecules, ions, electrons or formula units.

For 1 mol of CaCl₂, there is 1 mol of Ca²⁺ ions and 2 mol of Cl⁻ ions. The coefficient or subscript tells you exactly what entity count changes.

Use c = n/V with V in dm³. If the volume is given in cm³, divide by 1000 before you substitute; 25.0 cm³ is 0.0250 dm³.

I usually write the volume conversion on its own line. It keeps the chemistry visible and stops a perfectly good mole calculation being spoiled by a factor of 1000.

Use N = nNₐ to go from moles to particles and n = N/Nₐ to come back. Decide what the particles are before you calculate: atoms, molecules, ions, electrons or formula units.

If the question asks for atoms inside molecules, find the number of molecules first and then multiply by the number of those atoms in each molecule. Do not change particle type halfway through without accounting for it.

Mass and amount are linked by n = m/M, where M is molar mass in g mol⁻¹. The numerical value of M matches Mᵣ or relative formula mass, but Mᵣ itself has no unit.

Before you calculate, decide which quantity is unknown and rearrange symbolically. Writing m = nM or M = m/n first is safer than trying to remember whether to multiply or divide.

For a solution calculation, I’d put moles in the middle: use n = cV with V in dm³, apply the balanced-equation ratio, then convert the target moles into the concentration or volume the question wants.

That moles-in-the-middle route works even when the two concentrations look temptingly similar. Only skip the stoichiometric step if the balanced equation really gives a 1:1 ratio.

For pV = nRT, convert before substituting: p in Pa, V in m³ and T in K. Useful conversions are kPa × 1000, cm³ ÷ 10⁶ and °C + 273.

Write the converted values beside the equation. The numerical value of R does not need to be memorised; use the value supplied and make your units consistent with it.

Rearrange pV = nRT for the unknown before putting numbers in. Once we’ve written, for example, V = nRT/p or n = pV/RT, it is much easier to see which converted quantity belongs in each position.

If the answer is a gas volume, check its scale after converting back to the requested unit. A result of 1.2 × 10⁻⁵ m³ is 12 cm³, not 12 000 cm³.

An empirical formula is the simplest whole-number ratio of atoms of each element. It may be the same as the molecular formula, but it does not have to be.

Multiplying every subscript by the same whole number leaves the empirical ratio unchanged: CH₂O, C₂H₄O₂ and C₆H₁₂O₆ all reduce to 1:2:1. That is why empirical formula alone does not tell you molecular size.

To decide what a molecular formula tells you, think actual atom counts in one molecule. It must be a whole-number multiple of the empirical formula.

A molecular formula must be compatible with the empirical ratio. If the empirical formula is CH₂ and Mᵣ shows a multiplier of 3, the molecular formula is C₃H₆.

Find the empirical-formula mass, then calculate multiplier = Mᵣ ÷ empirical-formula mass. The multiplier should be a whole number; multiply every subscript in the empirical formula by it.

If the multiplier is nowhere near an integer, do not force it. Go back to the empirical ratio or the Mᵣ you used and find the earlier error.

Turn each mass or percentage into moles by dividing by Aᵣ, then divide every mole value by the smallest. Convert the ratio to small whole numbers before writing the empirical formula.

Ratios such as 1 : 1.5 are not rounding errors: multiply the whole ratio by 2 to get 2 : 3. With percentage data, treating the sample as 100 g makes the first step especially quick.

Once you have the empirical formula, calculate its formula mass and divide the molecular Mr by it. That gives the integer multiplier for the molecular formula.

The multiplier should be a sensible whole number. If Mᵣ/(empirical-formula mass) comes out nowhere near an integer, check the empirical formula, relative masses and any rounding before multiplying the subscripts.

When you move from a full equation to an ionic equation, strip out spectator ions and keep only the species that actually change. Balance the atoms first, then check that the total charge is the same on both sides.

For an ionic equation, split strong aqueous electrolytes into ions, then cancel spectators. Do not split solids, liquids, gases or weak molecular species merely because ions appear elsewhere in the reaction.

Use atom economy = Mr of desired product ÷ total Mr of reactants from the balanced equation × 100. Include stoichiometric coefficients when finding the total reactant mass represented by the equation.

Use the stoichiometric quantities from the balanced equation. If two moles of a reactant are required, its contribution to the denominator is 2Mᵣ, not one Mᵣ.

To judge atom economy, connect the percentage to where the reactant atoms finish up. High atom economy means more of them end up in the desired product, so there is generally less waste to separate, treat or dispose of and less material is bought for unwanted products.

Atom economy is only one process metric. A route can have excellent atom economy yet still use hazardous reagents, much energy or problematic solvents, so evaluate the whole process. One percentage cannot answer the sustainability question on its own.

Write chemically correct formulae first, then balance by changing coefficients only. Once the atom counts match, add or check state symbols if the question requires them.

Before you move on, count each element on both sides. For ionic equations, check total charge as well as atoms; a balanced-looking equation can still be electrically wrong.

When the unfamiliar reactants and products are supplied, treat balancing as a conservation problem. The products have already been supplied. Change coefficients only and leave a complicated species until later if that makes the count easier.

Finish by checking every element and, for ionic equations, the total charge. Never repair an imbalance by changing a subscript inside a supplied formula.

Whatever the starting data—mass, gas volume or solution concentration—convert it to moles before using the balanced-equation ratio. The coefficients tell you the mole relationship between the reacting species.

After the mole ratio, convert the target amount into the unit actually asked for: mass, concentration, gas volume, percentage yield or another quantity. Keeping those three stages separate makes multi-step questions much easier to audit.