Atomic structure and isotopes

In a nutshell

Every atom is built from protons, neutrons and electrons. The numbers of each decide which element it is, which isotope it is and what charge it carries.

Single atoms are far too light to weigh in grams, so chemists compare their masses with one twelfth of the mass of an atom of carbon-12. This subtopic sets up that relative mass scale and shows you how to read a mass spectrum and calculate a relative atomic mass from it.

Assumed knowledge: GCSE atomic structure (protons, neutrons, electrons, atomic number and mass number).

Core content

Atomic structure and isotopes

The three particles in an atom

Nearly all of an atom's mass is in the nucleus, because protons and neutrons are each about 1836 times heavier than an electron.

Atomic number and mass number

• Atomic number, Z: the number of protons in the nucleus. It fixes which element the atom is.
• Mass number, A: the total number of protons and neutrons in the nucleus.

Both numbers are shown in nuclide notation, with the mass number on top and the atomic number underneath, for example $^{56}_{26}\mathrm{Fe}$ (iron-56).

Counting the particles in atoms and ions

• A positive ion has lost electrons: Fe³⁺ has 26 - 3 = 23 electrons.
• A negative ion has gained electrons: S^2- has 16 + 2 = 18 electrons.
• Forming an ion changes only the electrons. The numbers of protons and neutrons stay the same.

Isotopes

Isotopes are atoms of the same element with different numbers of neutrons and different masses.

"Same element" means the same number of protons, so the atomic number is the same and only the neutron number (and so the mass number) changes.

Isotopes of an element have the same number of electrons arranged in the same way, so they react in the same way.

Models of the atom change with evidence

The picture above (a tiny nucleus with electrons in shells) is a model. Different models explain different things: the Bohr model, with electrons in fixed shells, is good at explaining patterns in the periodic table.

Models are accepted, changed or rejected as new evidence appears:

Relative mass

The carbon-12 standard

All relative masses are measured against the same standard: one twelfth of the mass of an atom of carbon-12. On this scale a carbon-12 atom has a mass of exactly 12.

Relative isotopic mass: the mass of an atom of an isotope compared with 1/12th of the mass of an atom of carbon-12.

Relative atomic mass, A_r: the weighted mean mass of an atom of an element compared with 1/12th of the mass of an atom of carbon-12.

The difference between them:

• Relative isotopic mass is for one isotope (for example chlorine-37 has a relative isotopic mass of 37).
• Relative atomic mass is for the element: a weighted mean of all its isotopes, taking their abundances into account (chlorine is 35.5).
• Both are ratios of two masses, so they have no units.

Still don't get it? · relative mass and the carbon-12 standard

Imagine you have to compare the weights of lots of different sweets, but your scales only read in kilograms, so every sweet reads as "0". A smarter trick: pick one sweet as the reference, cut it into 12 equal slices, and say how many slices each other sweet weighs. You never need the real weight, only the comparison.

Now step by step:

1. One atom weighs roughly 2 × 10^-23 g. Numbers that small are awkward to use.
2. So chemists choose one reference atom, carbon-12, and give it a mass of exactly 12.
3. One twelfth of a carbon-12 atom is then 1 unit on the scale. A proton or a neutron weighs about 1 unit.
4. Any other atom's relative mass is "how many of those twelfths it weighs". Because it is one mass divided by another, the units cancel and the answer has no units.
5. Do it for a single isotope and you get the relative isotopic mass. Do it for the element, averaging over all its isotopes by how common each is, and you get the relative atomic mass.

Exam wording: relative isotopic mass is the mass of an atom of an isotope compared with 1/12th of the mass of an atom of carbon-12; relative atomic mass is the weighted mean mass of an atom of an element compared with 1/12th of the mass of an atom of carbon-12.

Reading a mass spectrum

Mass spectrometry finds the relative isotopic masses and relative abundances of the isotopes in a sample of an element. You do not need to know how the mass spectrometer works, only how to read and use its spectrum.

• The x-axis is m/z (mass-to-charge ratio). In this course every ion has a single positive charge (1+), so m/z = the relative isotopic mass.
• Each peak is one isotope: the number of peaks is the number of isotopes.
• The height of a peak is the relative abundance of that isotope. It may be given as a percentage, or relative to the tallest peak set at 100.

From this spectrum you can read that magnesium has three isotopes, with relative isotopic masses 24, 25 and 26, and percentage abundances 79.0%, 10.0% and 11.0%.

Calculating relative atomic mass from a spectrum

A_r is a weighted mean: multiply each relative isotopic mass by its abundance, add them up, then divide by the total abundance.

$A_r = \frac{\sum (\text{relative isotopic mass} \times \text{abundance})}{\sum \text{abundance}}$

• If the abundances are percentages, the total is 100.
• If they are relative abundances (for example the tallest peak set to 100), divide by their sum, not by 100.

Quick check: A_r must lie between the lightest and heaviest isotopic masses, and closest to the most abundant isotope.

Still don't get it? · why relative atomic mass is a weighted mean

Think of a class where 3 students score 10 in a quiz and 1 student scores 2. If you just average the two different scores, (10 + 2) ÷ 2 = 6, you get a number that describes nobody well. Most of the class scored 10. The fair average counts each score as many times as it happened: (3 × 10 + 1 × 2) ÷ 4 = 8.

Atoms work the same way:

1. Natural chlorine is a mixture of chlorine-35 and chlorine-37 atoms.
2. They are not present in equal amounts: about 3 in every 4 chlorine atoms are chlorine-35.
3. A plain average of 35 and 37 gives 36, which ignores how common each isotope is.
4. Counting each mass by how often it occurs: (75 × 35 + 25 × 37) ÷ 100 = 35.5.
5. That 35.5 is why chlorine's relative atomic mass is not a whole number, even though no single chlorine atom has a mass of 35.5.

Exam wording: relative atomic mass is the weighted mean mass of an atom of an element compared with 1/12th of the mass of an atom of carbon-12. The word weighted is what the calculation is doing.

Relative molecular mass and relative formula mass

Add up the relative atomic masses of every atom in the formula. Use A_r values from the Periodic Table on the data sheet.

• For simple molecules (for example H₂O, CO₂), the total is called the relative molecular mass, M_r.
• For compounds with giant structures (giant ionic like NaCl, or giant covalent like SiO₂), the total is called the relative formula mass.

You do not need to learn definitions of these two terms, but you must use the right one and calculate it. Like A_r, they have no units.

Worked examples

1. Protons, neutrons and electrons in atoms and ions

Find the numbers of protons, neutrons and electrons in $^{56}_{26}\mathrm{Fe}^{3+}$, $^{32}_{16}\mathrm{S}^{2-}$ and a hydroxide ion, OH^-, made from oxygen-16 and hydrogen-1.

Iron ion, Fe³⁺ (Z = 26, A = 56, charge 3+):

$\text{protons} = Z = 26$ $\text{neutrons} = A - Z = 56 - 26 = 30$ $\text{electrons} = 26 - 3 = 23$

Sulfide ion, S^2- (Z = 16, A = 32, charge 2-):

$\text{protons} = 16, \quad \text{neutrons} = 32 - 16 = 16$ $\text{electrons} = 16 - (-2) = 18$

Hydroxide ion, OH^- (oxygen-16, Z = 8; hydrogen-1, Z = 1): add the particles of each atom, then apply the charge to the whole ion.

$\text{protons} = 8 + 1 = 9$ $\text{neutrons} = (16 - 8) + (1 - 1) = 8$ $\text{electrons} = 9 + 1 = 10$

Where marks go: the common slip is adding electrons for a positive ion (writing 29 for Fe³⁺) or changing the number of protons. Only electrons change when an ion forms.

2. Relative atomic mass from percentage abundances

Use the magnesium mass spectrum in Core content (m/z 24, 25, 26 at 79.0%, 10.0%, 11.0%) to calculate the relative atomic mass of magnesium to 3 significant figures.

Multiply each relative isotopic mass by its percentage abundance:

$24 \times 79.0 = 1896$ $25 \times 10.0 = 250$ $26 \times 11.0 = 286$

Add them and divide by the total abundance (100, because these are percentages):

$A_r = \frac{1896 + 250 + 286}{100} = \frac{2432}{100} = 24.32$

$A_r = 24.3 \text{ (3 s.f., no units)}$

Check: 24.3 lies between 24 and 26 and is close to 24, the most abundant isotope.

3. Relative atomic mass when abundances do not add up to 100

A mass spectrum of boron shows two peaks: m/z = 10 with relative abundance 24.8, and m/z = 11 with relative abundance 100. Calculate the relative atomic mass of boron to 3 significant figures.

The tallest peak has been set to 100, so the abundances add up to 124.8, not 100.

$\text{total abundance} = 24.8 + 100 = 124.8$

$A_r = \frac{(10 \times 24.8) + (11 \times 100)}{124.8} = \frac{248 + 1100}{124.8} = \frac{1348}{124.8} = 10.80$

$A_r = 10.8 \text{ (3 s.f., no units)}$

Where marks go: dividing by 100 instead of 124.8 gives 13.5, which is impossible because it is heavier than the heaviest isotope (11). Always run the "between the isotopes" check.

4. Percentage abundances from a relative atomic mass

Chlorine has two isotopes, chlorine-35 and chlorine-37. Its relative atomic mass is 35.5. Calculate the percentage abundance of each isotope.

Let the fraction of chlorine-35 be $x$. The fraction of chlorine-37 is then $(1 - x)$.

$35x + 37(1 - x) = 35.5$ $35x + 37 - 37x = 35.5$ $37 - 2x = 35.5$ $2x = 1.5$ $x = 0.75$

Chlorine-35 = 75.0% and chlorine-37 = 100 - 75.0 = 25.0%.

Check: (35 × 75.0 + 37 × 25.0) ÷ 100 = 35.5, which matches the data.

5. Relative molecular mass and relative formula mass

Calculate (a) the relative molecular mass of carbon dioxide, CO₂, and (b) the relative formula mass of calcium nitrate, Ca(NO₃)₂. Use A_r values from the data sheet: C 12.0, O 16.0, Ca 40.1, N 14.0.

(a) CO₂ is a simple molecule, so this is a relative molecular mass:

$M_r = 12.0 + (2 \times 16.0) = 44.0$

(b) Calcium nitrate is ionic (a giant structure), so this is a relative formula mass. The subscript 2 after the bracket multiplies everything inside it: 2 N and 6 O.

$\text{relative formula mass} = 40.1 + (2 \times 14.0) + (6 \times 16.0)$ $= 40.1 + 28.0 + 96.0 = 164.1$

Where marks go: forgetting to double the N (giving 40.1 + 14.0 + 96.0 = 150.1) is the classic bracket slip. Give the answer with no units.

Common exam mistakes

• Defining isotopes as "elements with different numbers of neutrons". Isotopes are atoms of the same element. Saying they have different numbers of protons (or a different atomic number) is wrong: that would make them different elements.
• Leaving "weighted" out of the relative atomic mass definition. "The mean mass of an atom" is not enough; A_r is the weighted mean mass of an atom of an element.
• Dropping the "1/12th" or the "-12". Comparing with "the mass of carbon-12", or with "1/12th of the mass of carbon", is incomplete. The standard is 1/12th of the mass of an atom of carbon-12.
• Mixing one atom with one mole in a definition. "The mass of one mole of atoms compared with 1/12th of the mass of one atom of carbon-12" compares two different amounts. Keep both sides about an atom.
• Getting electrons in ions backwards. Fe³⁺ has 23 electrons, not 29; S^2- has 18, not 14. Positive ions have lost electrons, negative ions have gained them, and the proton number never changes.
• Dividing by 100 when the abundances do not total 100. Divide by the sum of the abundances. An A_r outside the range of the isotopic masses is a sure sign of this slip.
• Taking a simple average of the isotopic masses. Chlorine's A_r is 35.5, not 36: each mass must be weighted by its abundance.
• Giving A_r or M_r units. They are ratios, so they have no units. Writing "g" or "g mol^-1" confuses them with mass or molar mass.
• Calling the mass of NaCl a "relative molecular mass". NaCl has a giant ionic structure, not molecules, so use relative formula mass.
• Bracket slips in formula masses. In Ca(NO₃)₂ the 2 multiplies both N and O.

Key definitions

• Isotopes: atoms of the same element with different numbers of neutrons and different masses.
• Atomic number (Z): the number of protons in the nucleus of an atom.
• Mass number (A): the total number of protons and neutrons in the nucleus of an atom.
• Relative isotopic mass: the mass of an atom of an isotope compared with 1/12th of the mass of an atom of carbon-12.
• Relative atomic mass, A_r: the weighted mean mass of an atom of an element compared with 1/12th of the mass of an atom of carbon-12.
• Relative molecular mass, M_r (simple molecules) and relative formula mass (giant structures): the sum of the relative atomic masses of all the atoms in the formula. Definitions not required; know which term to use.

Specification

• I can state that isotopes are atoms of the same element with different numbers of neutrons and different masses. (2.1.1(a))
• I can work out the numbers of protons, neutrons and electrons in an atom or ion from its atomic number, mass number and ionic charge. (2.1.1(b))
• I can describe how accepted models of atomic structure have changed as new evidence appeared, and say what the Bohr model explains. (2.1.1(b), HSW1 and HSW7)
• I can define relative isotopic mass and relative atomic mass, both against 1/12th of the mass of an atom of carbon-12. (2.1.1(c))
• I can read relative isotopic masses and relative abundances from a mass spectrum of singly charged ions. (2.1.1(d)(i))
• I can calculate the relative atomic mass of an element from the relative abundances of its isotopes, as a weighted mean. (2.1.1(d)(ii))
• I can use the terms relative molecular mass (simple molecules) and relative formula mass (giant structures) correctly, and calculate them from relative atomic masses. (2.1.1(e))

Related notes

• Compounds, formulae and equations: writing the formulae you calculate masses for
• Amount of substance: relative masses become molar masses in g mol^-1
• Electron structure: how the electrons in each atom are arranged in shells and orbitals
• Analytical techniques: mass spectra of organic compounds and the molecular ion peak