C10: Relative Masses and Moles
Understanding relative atomic mass, relative formula mass, the mole concept and Avogadro's constant — the fundamental building blocks of quantitative chemistry.
Relative Atomic Mass (Ar)
Relative atomic mass (Ar) is the mean mass of an atom of an element compared to 1/12th the mass of a carbon-12 atom. It takes into account the abundance of each isotope of the element.
Ar values are not whole numbers because most elements exist as mixtures of isotopes. For example, chlorine has Ar = 35.5 because it is roughly 75% 35Cl and 25% 37Cl.
You can find Ar values on the periodic table. Common values to remember: H = 1, C = 12, N = 14, O = 16, Na = 23, Mg = 24, S = 32, Cl = 35.5, Ca = 40, Fe = 56.
Worked Example: Calculating Ar from Isotopic Abundance
A sample of copper contains 69% 63Cu and 31% 65Cu. Calculate the Ar of copper.
Ar = (69 × 63 + 31 × 65) / 100
Ar = (4347 + 2015) / 100
Ar = 6362 / 100 = 63.6
Relative Formula Mass (Mr)
Relative formula mass (Mr) is the sum of the relative atomic masses of all the atoms in a formula. For covalent molecules it is sometimes called relative molecular mass.
Mr = sum of all Ar values in the formula
To calculate Mr, write out the formula, list the Ar of each atom, multiply by the number of each atom present, then add them all together.
Worked Example: Mr of Mg(OH)2
Mg = 24, O = 16, H = 1
Mr = 24 + (2 × 16) + (2 × 1)
Mr = 24 + 32 + 2 = 58
Worked Example: Mr of Fe2(SO4)3
Fe = 56, S = 32, O = 16
Mr = (2 × 56) + (3 × 32) + (12 × 16)
Mr = 112 + 96 + 192 = 400
Always expand brackets in formulae carefully. Fe2(SO4)3 contains 3 sulfates, so that is 3 S and 3 × 4 = 12 O atoms.
The Mole Concept
A mole is the amount of substance that contains the same number of particles as there are atoms in 12 g of carbon-12. One mole of any substance always contains the same number of particles.
The mole is the SI unit for amount of substance. The symbol for mole is mol. Using moles allows chemists to count particles by weighing substances, since particles are far too small to count individually.
One mole of any substance has a mass in grams equal to its relative formula mass. For example, 1 mol of H2O (Mr = 18) has a mass of 18 g.
Number of moles = mass (g) / Mr
This equation can be rearranged to find mass or Mr:
mass = moles × Mr
Mr = mass / moles
Worked Example: Calculating Moles from Mass
How many moles are in 60 g of NaOH?
Mr of NaOH = 23 + 16 + 1 = 40
moles = mass / Mr = 60 / 40 = 1.5 mol
Worked Example: Calculating Mass from Moles
What is the mass of 0.25 mol of CaCO3?
Mr of CaCO3 = 40 + 12 + (3 × 16) = 100
mass = moles × Mr = 0.25 × 100 = 25 g
Worked Example: Finding Mr from Mass and Moles
A 0.2 mol sample of a compound has a mass of 13.2 g. Find its Mr.
Mr = mass / moles = 13.2 / 0.2 = 66
Avogadro's Constant
Avogadro's constant (NA) is the number of particles in one mole of a substance. Its value is 6.02 × 1023 per mole.
Number of particles = moles × 6.02 × 1023
This means that 1 mol of any substance always contains exactly 6.02 × 1023 particles. The particles could be atoms, molecules, ions or formula units depending on the substance.
| Substance | 1 mole mass | Number of particles | Type of particle |
|---|---|---|---|
| Carbon (C) | 12 g | 6.02 × 1023 | Atoms |
| Water (H2O) | 18 g | 6.02 × 1023 | Molecules |
| Sodium chloride (NaCl) | 58.5 g | 6.02 × 1023 | Formula units |
| Oxygen (O2) | 32 g | 6.02 × 1023 | Molecules |
Worked Example: Number of Molecules from Mass
How many molecules are in 36 g of H2O?
Mr of H2O = (2 × 1) + 16 = 18
moles = mass / Mr = 36 / 18 = 2 mol
number of molecules = 2 × 6.02 × 1023 = 1.20 × 1024 molecules
Worked Example: Mass from Number of Atoms
What is the mass of 3.01 × 1023 atoms of iron (Fe)?
moles = number of particles / NA = 3.01 × 1023 / 6.02 × 1023 = 0.5 mol
mass = moles × Ar = 0.5 × 56 = 28 g
When asked for number of atoms (not molecules) in a compound, multiply by the number of atoms per molecule. E.g. 1 mol of H2O contains 3 × 6.02 × 1023 atoms (2 H + 1 O per molecule).
Using Moles in Equations
Balanced symbol equations show the mole ratios between reactants and products. The big numbers in front of formulae tell you how many moles of each substance react or are produced.
For example, in the equation 2H2 + O2 → 2H2O, the ratio of H2:O2:H2O is 2:1:2. This means 2 mol of hydrogen react with 1 mol of oxygen to produce 2 mol of water.
Worked Example: Using Mole Ratios
In the reaction N2 + 3H2 → 2NH3, how many moles of NH3 are produced from 1.5 mol of N2?
Mole ratio N2:NH3 = 1:2
So moles of NH3 = 1.5 × 2 = 3 mol
Always check the equation is balanced before using mole ratios. The coefficients (big numbers) are what determine the ratio, not the subscripts within formulae.
Practice Questions
Calculate the Mr of Al2(SO4)3. (Al = 27, S = 32, O = 16)
Mr = (2 × 27) + (3 × 32) + (12 × 16) = 54 + 96 + 192 = 342
How many moles are in 8 g of CuO? (Cu = 64, O = 16)
Mr of CuO = 64 + 16 = 80
moles = 8 / 80 = 0.1 mol
How many atoms are in 0.5 mol of carbon?
number of atoms = 0.5 × 6.02 × 1023 = 3.01 × 1023 atoms
What mass of NaCl is needed to provide 0.4 mol? (Na = 23, Cl = 35.5)
Mr of NaCl = 23 + 35.5 = 58.5
mass = 0.4 × 58.5 = 23.4 g
A 0.15 mol sample of a gas has a mass of 4.8 g. Calculate its Mr.
Mr = mass / moles = 4.8 / 0.15 = 32
This suggests the gas could be O2.
Maths Skills
Calculating Ar and Mr
Calculating Ar from isotopic abundance uses a weighted mean: Ar = (fraction × mass) for each isotope, then add them together. For example, if an element has 80% of isotope-20 and 20% of isotope-22: Ar = 0.80 × 20 + 0.20 × 22 = 16 + 4.4 = 20.4.
Calculating Mr requires careful multiplication and addition. Always expand brackets in formulae: for Al2(SO4)3, that is 2 Al, 3 S, and 3 × 4 = 12 O atoms. Common errors include forgetting to multiply subscripted atoms inside brackets by the number outside.
Calculating Moles
The mole equation moles = mass / Mr can be rearranged. Use a formula triangle: mass on top, moles and Mr on the bottom. Cover the one you want to find. For example, to find mass: mass = moles × Mr.
Avogadro's constant is written in standard form as 6.02 × 1023. Standard form is essential here because the number of particles is enormous. For example, 2 mol of water contains 2 × 6.02 × 1023 = 1.204 × 1024 molecules.
When multiplying or dividing with standard form, multiply the coefficients and add the exponents for multiplication, or subtract them for division. Always check your final answer is in correct standard form (number between 1 and 10 × power of 10).
Significant Figures
In mole calculations, give your answer to the same number of significant figures as the least precise data in the question. For example, if masses are given to 2 significant figures (e.g. 6.0 g), give your final answer to 2 significant figures too. Intermediate calculations should use at least one more significant figure to avoid rounding errors.
Do not confuse significant figures with decimal places. The number 0.00450 has 3 significant figures (4, 5, 0) but 5 decimal places. Leading zeros are not significant; trailing zeros after a decimal point are significant.
Common Misconceptions
Mass and Moles
Wrong: One mole of any substance always has the same mass Correct: One mole always has the same number of particles (6.02 × 1023), but the mass depends on the substance. 1 mol of H2O = 18 g, 1 mol of NaCl = 58.5 g — the mass in grams equals the Mr
Wrong: Mr has units of grams Correct: Mr (relative formula mass) is a ratio and has no units. Molar mass (the mass of one mole) has units of g/mol and has the same numerical value as Mr
Wrong: Ar and Mr are the same thing Correct: Ar is the relative atomic mass of a single element. Mr is the relative formula mass of a compound — the sum of all Ar values in the formula
Wrong: A mole is a fixed mass like a kilogram Correct: A mole is a fixed number of particles (6.02 × 1023), not a fixed mass. The mass of 1 mol depends on the substance
6-Mark Extended Question
Explain and Illustrate the Mole Concept
Explain what a mole is and why it is useful in chemistry. Use examples to illustrate your answer.
A mole is the SI unit for amount of substance. One mole of any substance contains exactly 6.02 × 1023 particles (Avogadro's constant). This number was chosen so that 1 mol of any substance has a mass in grams equal to its relative formula mass. [2 marks]
The mole is useful because atoms and molecules are far too small to count individually. By using moles, chemists can count particles by weighing, since 1 mol of carbon-12 has a mass of exactly 12 g. For example, 1 mol of H2O (Mr = 18) has a mass of 18 g and contains 6.02 × 1023 molecules. [2 marks]
Moles link mass to particle number through the equation moles = mass / Mr. This allows chemists to calculate exact amounts of reactants needed. For example, to produce 2 mol of NaCl, you need exactly 2 mol of NaOH and 2 mol of HCl. Without the mole concept, it would be impossible to measure these proportions by mass alone. [2 marks]
AO3: Analyse and Evaluate
Comparing Moles in Different Substances
A student has three samples: Sample A contains 44 g of CO2 (Mr = 44), Sample B contains 36 g of H2O (Mr = 18), and Sample C contains 64 g of O2 (Mr = 32). Which sample contains the most moles? Which sample contains the most molecules? Explain your reasoning.
Moles of CO2 = 44 / 44 = 1.0 mol. Moles of H2O = 36 / 18 = 2.0 mol. Moles of O2 = 64 / 32 = 2.0 mol. Samples B and C both contain 2.0 mol, which is more than Sample A. Both B and C contain the same number of molecules (2 × 6.02 × 1023). This shows that mass alone does not tell you which substance has more particles — you must calculate moles using Mr. A heavier substance like O2 needs a greater mass to contain the same number of moles as a lighter substance like H2O.
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