Part 2: Relative masses and isotope calculations

Interpret singly charged isotope peaks and calculate weighted relative masses without confusing them with molar masses.

Carbon-12 is the reference

Relative isotopic mass is the mass of an atom of an isotope compared with one twelfth of the mass of a carbon-12 atom. Relative atomic mass, Ar, uses the weighted mean mass of atoms of an element relative to the same reference. Both are ratios and have no unit.

Mass number is an integer particle count. Relative isotopic mass is a measured mass ratio and need not be exactly an integer. Use the isotope masses supplied in a question. Relative molecular mass adds the Ar values in a molecule; for a giant ionic substance use relative formula mass instead.

Relative mass is a ratio, not a weighed amount

Choosing one twelfth of a carbon-12 atom as the reference gives a common scale for atoms and molecules. “Relative” means divide by that reference mass. The mass units cancel. A carbon-12 atom therefore has relative isotopic mass exactly 12 by definition, whereas its mass number is 12 because it contains six protons and six neutrons: identical numbers arise from different definitions.

Relative molecular mass compares the average mass of a molecule with that same reference. Relative formula mass applies the atom-ratio calculation to a formula unit when there is no discrete molecule, for example NaCl or SiO₂ in a giant structure. Neither quantity has g mol⁻¹ as its unit. Molar mass is a different quantity: mass divided by amount of substance.

Work from abundance, not the midpoint

A mass spectrum plots relative abundance against m/z. For the singly charged isotope ions in this part of OCR AS, z = 1, so the peaks identify the isotopic masses numerically. Instrument design and time-of-flight calculations are not required here.

Illustrative data: an element has isotopes of mass 10.0 and 11.0 with abundances 20.0% and 80.0%. Ar = (10.0 × 20.0 + 11.0 × 80.0)/100 = 10.8. The answer lies nearer 11 because that isotope is more abundant. If relative peak heights total 50 rather than 100, divide by 50.

Reverse example: isotopes 63 and 65 have mean 63.6. Let fraction f be isotope 63: 63f + 65(1 − f) = 63.6. Therefore 2f = 1.4 and f = 0.700: 70.0% of isotope 63, 30.0% of isotope 65.

Read a spectrum, then calculate the weighted mean

The illustrated spectrum uses invented singly charged isotope ions at m/z 20, 21 and 22 with relative intensities 90, 5 and 5. Multiply each mass by its own intensity: 20 × 90 = 1800; 21 × 5 = 105; 22 × 5 = 110. Add 2015 and divide by total intensity 100 to obtain Ar = 20.15. The result lies close to the dominant isotope at 20.

Peak height is proportional to abundance for this simple interpretation. A peak normalised to 100 is the reference tallest peak; that does not necessarily mean 100% of the atoms. With heights 100 and 25 at masses 50 and 52, divide by 125: Ar = (5000 + 1300)/125 = 50.4. Scaling every height by the same factor leaves the mean unchanged.

Swipe horizontally to view the whole diagram.

Find an unknown abundance without guessing

Suppose three isotope masses are 24, 25 and 26. The middle isotope has abundance 10.0% and Ar = 24.50. Let the percentage of mass-24 atoms be x; mass-26 abundance must then be 90.0 − x. The three percentages must sum to 100, not 110.

Write 24x + 25(10.0) + 26(90.0 − x) = 24.50 × 100. Expanding gives 2590 − 2x = 2450, so x = 70.0. The mixture is 70.0%, 10.0%, 20.0%. Substitution gives (1680 + 250 + 520)/100 = 24.50, verifying both the mean and the total.

Use a reasonableness check before accepting any result: percentages must be between 0 and 100 and a weighted mean must lie between the smallest and largest isotope masses. A negative percentage usually signals an algebra or data-pairing error.

Add every atom in the formula

For CO₂, Mr = 12.0 + 2(16.0) = 44.0. For Mg(NO₃)₂, relative formula mass = 24.3 + 2[14.0 + 3(16.0)] = 148.3. Brackets multiply the whole group. A molar mass has the same numerical value in g mol⁻¹ but describes mass per mole, not a dimensionless ratio.

Quick checks

Original Finesse questions. Reveal the indicative worked solutions after attempting each question; these are not official OCR A mark allocations.

Sources