S1.2 The nuclear atom

The tallest peak in a mass spectrum, from the most abundant ion, assigned a relative abundance of 100.

Evidence for the nuclear model

Models of the atom have changed as new evidence appeared. In the early nineteenth century John Dalton described atoms as tiny, indivisible solid spheres, different for each element. The discovery of the negatively charged electron in 1897 showed that atoms contain smaller particles, and J. J. Thomson proposed the "plum pudding" model: electrons embedded in a sphere of spread-out positive charge. Each later model kept what still fitted the evidence and replaced what did not.

The gold-foil experiment

In 1909 Hans Geiger and Ernest Marsden, working with Ernest Rutherford, fired a beam of alpha particles at a very thin sheet of gold foil. Alpha particles are helium nuclei, ${{\mathrm{He}{\vphantom{X}}^{2+}}}$: small, fast-moving and positively charged. A screen coated with zinc sulfide surrounded the foil and gave a tiny flash of light wherever an alpha particle struck it, so the deflections could be counted.

If the plum pudding model were correct, the positive charge and mass of each gold atom would be spread thinly through it, and every alpha particle would pass through with at most a slight deflection. The results were very different.

In 1911 Rutherford proposed the nuclear model: a tiny, dense, positively charged nucleus at the centre of the atom, with electrons moving in the space around it. From the fraction of particles scattered back he estimated that the nucleus is about 100 000 times smaller in diameter than the atom. If a hydrogen atom were enlarged to the size of a football stadium, its nucleus would be about the size of a pea at the centre spot.

The model has been refined since. In 1913 Niels Bohr proposed that the electrons occupy fixed energy levels, which explained the line spectra of elements; the arrangement of electrons is developed with electron configurations. In 1932 James Chadwick discovered the neutron, an uncharged particle in the nucleus, which accounted for the fact that nuclei have about twice the mass that their protons alone could provide. The history of atomic models shows how scientific models are tested against evidence and revised when they fail.

Protons, neutrons and electrons

Atoms are made of three kinds of subatomic particle. The protons and neutrons are in the nucleus, and together they are the nucleons. The electrons occupy the space around the nucleus. Because the masses and charges are tiny in ordinary units, chemists compare them using relative values.

The actual values are listed in the data booklet: the proton has a mass of $1.672622\times10^{-27}\ \text{kg}$, the neutron $1.674927\times10^{-27}\ \text{kg}$ and the electron $9.109384\times10^{-31}\ \text{kg}$. Dividing the proton mass by the electron mass gives about 1836, so the mass of an atom is almost entirely the mass of its nucleus and the electrons can be ignored when comparing masses. The charge on a proton is $+1.602177\times10^{-19}\ \text{C}$ and on an electron exactly the same size but negative.

An atom is electrically neutral because it contains equal numbers of protons and electrons. The electrostatic attraction between the positive nucleus and the negative electrons holds the electrons in the atom. The positive protons in the nucleus repel each other, but a much stronger, very short-range force, the strong nuclear force, holds the nucleons together.

The chemical behaviour of an atom depends on its electrons, particularly those in the outer energy level, because these are the electrons that are transferred or shared when atoms react. The number of electrons in a neutral atom is fixed by the number of protons, which is why the number of protons determines the chemistry of an element and its position in the periodic table.

Atomic number, mass number and the nuclear symbol

The atomic number identifies the element: every atom with 6 protons is carbon and every atom with 17 protons is chlorine. Elements are arranged in the periodic table in order of increasing atomic number, so the atomic number of an element is also its position in the table. The mass number is always a whole number, because it counts particles. Since the electrons contribute almost nothing to the mass, the mass number is close to the mass of the atom in relative units.

An atom or ion is described by its nuclear symbol, $^{A}_{Z}\text{X}$, where X is the symbol of the element. The mass number is written at the top left, the atomic number at the bottom left and any charge at the top right.

The numbers of each particle follow directly from the symbol:

• protons $= Z$

• neutrons $= A - Z$

• electrons $= Z - \text{charge}$ (for a neutral atom, electrons $= Z$)

An ion is a charged particle formed when an atom gains or loses electrons. A positive ion, or cation, has lost electrons, so it has fewer electrons than protons: ${{\mathrm{Mg}{\vphantom{X}}^{2+}}}$ has 12 protons and 10 electrons. A negative ion, or anion, has gained electrons: ${{\mathrm{O}{\vphantom{X}}^{2-}}}$ has 8 protons and 10 electrons. The number of protons never changes when an ion forms; only nuclear reactions change the nucleus.

Species with the same number of electrons are isoelectronic. The ions ${{\mathrm{O}{\vphantom{X}}^{2-}}}$, ${{\mathrm{F}{\vphantom{X}}^{-}}}$, ${{\mathrm{Na}{\vphantom{X}}^{+}}}$ and ${{\mathrm{Mg}{\vphantom{X}}^{2+}}}$ and the atom ${{\mathrm{Ne}}}$ all have 10 electrons, but different numbers of protons, so they are different species with different sizes and properties.

Comparing species

Many problems compare two or more species and ask which quantity they share or how two counts differ. The reliable method is to write out all three counts for every species before comparing them. Two species with the same mass number are not necessarily alike: tellurium-128, ${{{\vphantom{X}}^{\hphantom{128}}_{\hphantom{52}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{128}}_{\vphantom{2}\mathllap{\smash[t]{52}}}\mathrm{Te}}}$, and the iodide ion ${{{\vphantom{X}}^{\hphantom{128}}_{\hphantom{53}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{128}}_{\vphantom{2}\mathllap{\smash[t]{53}}}\mathrm{I}{\vphantom{X}}^{-}}}$ have the same number of nucleons but different numbers of protons (52 and 53), neutrons (76 and 75) and electrons (52 and 54). Species with the same number of protons belong to the same element, whatever their charge or mass number.

Isotopes

Most elements exist naturally as a mixture of isotopes. An isotope is named by its element and mass number, as in chlorine-35 and chlorine-37, or written ${{{\vphantom{X}}^{\hphantom{35}}_{\hphantom{}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{35}}_{\vphantom{2}\mathllap{\smash[t]{}}}\mathrm{Cl}}}$ and ${{{\vphantom{X}}^{\hphantom{37}}_{\hphantom{}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{37}}_{\vphantom{2}\mathllap{\smash[t]{}}}\mathrm{Cl}}}$; the atomic number can be left out because the symbol already fixes it. Hydrogen has three isotopes, with 0, 1 and 2 neutrons; the heavier two are called deuterium and tritium.

Abundances: IUPAC representative isotopic compositions (2013).

Chemical properties of isotopes

Isotopes of an element have the same chemical properties. Chemical reactions involve electrons, and isotopes have the same number of protons and therefore the same number and arrangement of electrons. They also have the same nuclear charge, so their electrons are attracted to the nucleus equally strongly. The extra neutrons carry no charge and do not affect this attraction, so the first ionization energy of chlorine-37 is the same as that of chlorine-35, and both form chloride ions, ${{\mathrm{Cl}{\vphantom{X}}^{-}}}$, in exactly the same reactions.

Physical properties of isotopes

Physical properties that depend on the mass of the particles are different for different isotopes. Heavier isotopes, and compounds containing them, have higher densities, slightly higher melting and boiling points, and diffuse more slowly. The differences are largest for hydrogen, where deuterium has twice the mass of hydrogen-1.

Ice made of heavy water is denser than ordinary liquid water, so a heavy-water ice cube sinks in a glass of ordinary water, whereas ordinary ice floats. The chemistry of the two waters is the same: both react with sodium to give hydrogen (or deuterium) and both dissolve salts. The difference in mass is also used on an industrial scale. Natural uranium contains only about 0.7% uranium-235, the isotope used in nuclear reactors; gas centrifuges spin gaseous uranium hexafluoride, ${{\mathrm{UF}{\vphantom{X}}_{\smash[t]{6}}}}$, so that the slightly heavier ${{{\vphantom{X}}^{\hphantom{238}}_{\hphantom{}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{238}}_{\vphantom{2}\mathllap{\smash[t]{}}}\mathrm{UF}{\vphantom{X}}_{\smash[t]{6}}}}$ molecules move outwards and the gas left near the centre is enriched in uranium-235.

Some isotopes have unstable nuclei that break down, emitting radiation; they are radioisotopes. Carbon-14, for example, decays with a half-life of about 5730 years, and the fraction remaining in once-living material is used to date it. Because isotopes behave identically in reactions but can be detected by their mass or radioactivity, they are used as tracers. When ethanol containing oxygen-18 is esterified with ethanoic acid, all of the oxygen-18 ends up in the ester and none in the water, showing which bonds break in the reaction mechanism.

The reason mass matters for physical properties follows from the kinetic molecular theory. At the same temperature all particles have the same average kinetic energy, $\tfrac{1}{2}mv^2$, so heavier particles move more slowly on average. Gaseous ${{{\vphantom{X}}^{\hphantom{238}}_{\hphantom{}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{238}}_{\vphantom{2}\mathllap{\smash[t]{}}}\mathrm{UF}{\vphantom{X}}_{\smash[t]{6}}}}$ therefore diffuses slightly more slowly than ${{{\vphantom{X}}^{\hphantom{235}}_{\hphantom{}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{235}}_{\vphantom{2}\mathllap{\smash[t]{}}}\mathrm{UF}{\vphantom{X}}_{\smash[t]{6}}}}$, and ${{\mathrm{D}{\vphantom{X}}_{\smash[t]{2}}}}$ diffuses more slowly than ${{\mathrm{H}{\vphantom{X}}_{\smash[t]{2}}}}$. Heavier molecules also pack more mass into the same volume, which raises the density, and they need slightly more energy to escape from the liquid, which raises the boiling point. The relative molecular mass of a compound changes too when one of its elements is enriched in a heavier isotope, which is how isotope-labelled compounds are recognized in a mass spectrometer.

Relative atomic mass

The masses of atoms are compared on a scale on which one atom of carbon-12 has a mass of exactly 12 units. The mass of a single isotope on this scale is its relative isotopic mass, which is very close to its mass number: 34.97 for chlorine-35 and 36.97 for chlorine-37. A natural sample of an element contains a mixture of isotopes, so its atoms do not all have the same mass, and the relative atomic mass is an average.

The mean is weighted because the isotopes are not present in equal numbers. In a natural sample of chlorine about 76 atoms in every 100 are chlorine-35 and 24 are chlorine-37, so the average lies much closer to 35 than to 37. The grid of one hundred chlorine atoms makes the weighting visible.

The relative atomic mass is found by multiplying the mass of each isotope by its percentage abundance, adding the results and dividing by 100:

If the abundances are given as fractions or decimals that add up to 1, the division by 100 is left out. Relative atomic masses are usually not whole numbers, because they are averages of isotopes of different mass. The values in the periodic table of the data booklet are given to two decimal places.

When only mass numbers are given, they are used as the isotopic masses. For boron, which is $19.9\%$ boron-10 and $80.1\%$ boron-11, this gives $A_\text{r} = (10 \times 19.9 + 11 \times 80.1)/100 = 10.80$, close to the data booklet value of 10.81. The small difference arises because relative isotopic masses are not exactly whole numbers.

Finding isotopic abundances from the relative atomic mass

For an element with two isotopes, the calculation can be reversed to find their abundances. If one isotope has an abundance of $x\%$, the other must have an abundance of $(100 - x)\%$, so there is only one unknown.

Measuring isotope ratios

Isotope ratios are measured with a mass spectrometer, which detects the isotopes of an element and their relative abundances. The proportions of isotopes vary slightly between natural sources: the ratio of carbon-13 to carbon-12 differs between plants that photosynthesize in different ways, and the ratio of nitrogen-15 to nitrogen-14 in the atmosphere differs from that in fertilizers. Food laboratories use these small differences to detect fraud. Sugar cane and maize, for example, contain a slightly higher proportion of carbon-13 than the flowering plants that bees visit, so honey diluted with cheap cane or corn syrup has a measurably higher carbon-13 to carbon-12 ratio.

Because isotope ratios vary between sources, IUPAC gives the relative atomic masses of some elements, such as hydrogen, carbon and chlorine, as a small range rather than a single value; the data booklet gives one representative value for each.

Mass spectra of elements (HL)

A mass spectrometer turns atoms of the sample into positive ions, separates the ions by their mass-to-charge ratio, and counts how many arrive at each value of $m/z$. The output is a mass spectrum: a bar chart of relative abundance (or intensity) on the vertical axis against $m/z$ on the horizontal axis.

Almost all the ions formed carry a single positive charge ($z = 1$), so the $m/z$ value of each peak equals the mass of the isotope. Each peak in the mass spectrum of an element is therefore one isotope, and the height of the peak is proportional to its relative abundance. The mass spectrum of magnesium has three peaks, showing three isotopes.

A small fraction of atoms lose two electrons and form ${{\mathrm{X}{\vphantom{X}}^{2+}}}$ ions. With $z = 2$, these appear at half the mass of the isotope: magnesium-24 gives a very small peak at $m/z = 12$. Peaks from doubly charged ions are much smaller than the main peaks and are left out when the relative atomic mass is calculated.

Calculating the relative atomic mass from a spectrum

When the peak heights are percentages, the calculation is the same as for abundance data. Often the tallest peak, the base peak, is set to 100 and the others are given relative to it, as in the spectrum of magnesium. The heights must then be divided by their total, not by 100.

Identifying an element from its spectrum

Because each element has a characteristic set of isotopes, a mass spectrum identifies the element. The relative atomic mass calculated from the spectrum is compared with the values in the periodic table: a spectrum with peaks at 63 and 65 in a ratio of about 7 : 3 gives $A_\text{r} \approx 63.6$, which identifies copper ($A_\text{r} = 63.55$). Using mass numbers rather than exact isotopic masses shifts the result by a few hundredths, so the calculated value is matched to the nearest element, and the number of peaks and their positions confirm the match.

Elements that form molecules

An element that exists as diatomic molecules, such as chlorine or bromine, also gives peaks for molecular ions, ${{\mathrm{X}{\vphantom{X}}_{\smash[t]{2}}{\vphantom{X}}^{+}}}$, formed when a whole molecule loses one electron. In the mass spectrum of chlorine the peaks at $m/z$ 35 and 37 come from ${{{\vphantom{X}}^{\hphantom{35}}_{\hphantom{}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{35}}_{\vphantom{2}\mathllap{\smash[t]{}}}\mathrm{Cl}{\vphantom{X}}^{+}}}$ and ${{{\vphantom{X}}^{\hphantom{37}}_{\hphantom{}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{37}}_{\vphantom{2}\mathllap{\smash[t]{}}}\mathrm{Cl}{\vphantom{X}}^{+}}}$ ions; the peaks at 70, 72 and 74 come from ${{\mathrm{Cl}{\vphantom{X}}_{\smash[t]{2}}{\vphantom{X}}^{+}}}$ ions made of two chlorine-35 atoms, one of each isotope, and two chlorine-37 atoms.

The heights of the molecular-ion peaks follow from the probability of each combination. About three quarters of chlorine atoms are chlorine-35, so the chance that both atoms in a molecule are chlorine-35 is $\tfrac{3}{4}\times\tfrac{3}{4} = \tfrac{9}{16}$. The chance of one of each is $2\times\tfrac{3}{4}\times\tfrac{1}{4} = \tfrac{6}{16}$ (the factor 2 because either atom can be the chlorine-37), and of two chlorine-37 atoms $\tfrac{1}{4}\times\tfrac{1}{4} = \tfrac{1}{16}$. The peaks at 70, 72 and 74 are therefore in the ratio 9 : 6 : 1. For bromine, whose two isotopes are about equally abundant, the ${{\mathrm{Br}{\vphantom{X}}_{\smash[t]{2}}{\vphantom{X}}^{+}}}$ peaks at 158, 160 and 162 are in the ratio 1 : 2 : 1.

Molecules of compounds also break into fragments in the mass spectrometer, and the pattern of fragment peaks is used to work out the structure of organic compounds; that interpretation is developed with the spectroscopic identification of organic compounds.

Applying the ideas in exam questions

Paper 1A: particles in an ion

Paper 1A: properties of isotopes

Paper 1A: estimating abundances

Paper 2: isotopes of copper

Paper 2: evidence for the nucleus

Paper 1B: lithium from two sources

A student uses an isotope database and a supplier's certificate to compare two samples of lithium. Each abundance has an uncertainty of $\pm 0.1\%$. Relative isotopic masses: lithium-6, 6.015; lithium-7, 7.016.

Paper 2: the mass spectrum of lead (HL)

Paper 1A: a two-peak spectrum (HL)

Self-check

• To do:

  I can describe the gold-foil experiment and explain how its results support the nuclear model.

• To do:

  I can explain how models of the atom have changed in response to new evidence.

• To do:

  I can state the relative masses, relative charges and locations of protons, neutrons and electrons.

• To do:

  I can define atomic number and mass number.

• To do:

  I can write and interpret the nuclear symbol of an atom or ion.

• To do:

  I can deduce the numbers of protons, neutrons and electrons in atoms and ions.

• To do:

  I can compare species by their numbers of protons, neutrons and electrons, including isotopes and isoelectronic species.

• To do:

  I can define isotopes and give examples.

• To do:

  I can explain why isotopes have the same chemical properties but different physical properties.

• To do:

  I can define relative atomic mass and explain why it is usually not a whole number.

• To do:

  I can calculate a relative atomic mass from isotope masses and abundances.

• To do:

  I can calculate the abundances of two isotopes from a relative atomic mass, and estimate them without a calculator.

• To do:

  I can state that mass spectrometry is used to measure isotope abundances and isotope ratios.

• To do:

  I can interpret the mass spectrum of an element in terms of its isotopes and their relative abundances (HL).

• To do:

  I can calculate a relative atomic mass from a mass spectrum, including when peak heights are relative to the tallest peak (HL).

• To do:

  I can identify an element from the peaks in its mass spectrum (HL).

• To do:

  I can explain the molecular-ion peaks in the mass spectrum of a diatomic element such as chlorine or bromine and calculate a relative molecular mass from them (HL).

1. A

   Protons: 27; neutrons: 32; electrons: 24

2. B

   Protons: 27; neutrons: 32; electrons: 30

3. C

   Protons: 27; neutrons: 59; electrons: 24

4. D

   Protons: 32; neutrons: 27; electrons: 24

(b) Protons $= 24$; neutrons $= 52 - 24 = 28$; electrons $= 24 - 3 = 21$ (three electrons lost).

(c) 35 protons means $Z = 35$, which is bromine. $A = 35 + 46 = 81$. There is one more electron than protons, so the charge is 1−:

(a) ${{{\vphantom{X}}^{\hphantom{40}}_{\hphantom{19}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{40}}_{\vphantom{2}\mathllap{\smash[t]{19}}}\mathrm{K}{\vphantom{X}}^{+}}}$ and ${{{\vphantom{X}}^{\hphantom{39}}_{\hphantom{19}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{39}}_{\vphantom{2}\mathllap{\smash[t]{19}}}\mathrm{K}}}$ both have 19 protons, so both are potassium, with 21 and 20 neutrons. ${{{\vphantom{X}}^{\hphantom{40}}_{\hphantom{20}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{40}}_{\vphantom{2}\mathllap{\smash[t]{20}}}\mathrm{Ca}}}$ has the same mass number as potassium-40 but is a different element.

(b) Neutrons minus electrons: $21 - 18 = 3$ for ${{\mathrm{K}{\vphantom{X}}^{+}}}$, $0$ for ${{\mathrm{Ca}}}$, $1$ for ${{{\vphantom{X}}^{\hphantom{39}}_{\hphantom{}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{39}}_{\vphantom{2}\mathllap{\smash[t]{}}}\mathrm{K}}}$ and $20 - 18 = 2$ for ${{\mathrm{Cl}{\vphantom{X}}^{-}}}$. The answer is ${{{\vphantom{X}}^{\hphantom{37}}_{\hphantom{17}}{\vphantom{X}}^{\smash[t]{\vphantom{2}}\mathllap{37}}_{\vphantom{2}\mathllap{\smash[t]{17}}}\mathrm{Cl}{\vphantom{X}}^{-}}}$.

(c) Both have 18 electrons: they are isoelectronic. Their numbers of protons and neutrons differ.

For natural ${{\mathrm{N}{\vphantom{X}}_{\smash[t]{2}}\mathrm{O}}}$, using $A_\text{r}({{\mathrm{N}}}) = 14.01$ and $A_\text{r}({{\mathrm{O}}}) = 16.00$: $M_\text{r} = (2 \times 14.01) + 16.00 = 44.02$. The enriched molecules are about 2 units heavier.

(b) The boiling point is slightly higher and the rate of diffusion lower, because both depend on the mass of the molecules. The reaction with copper is the same, because the nitrogen-15 atoms have the same number of protons and the same electron configuration as nitrogen-14 atoms, so the bonds form and break in the same way.

(b) Total of the heights: $51.4 + 100 + 48.6 = 200.0$.

This is close to $2 \times 79.90 = 159.80$ from the data booklet; the difference again comes from using mass numbers.

(c) Bromine has only two isotopes, of mass 79 and 81, so the only possible totals for two atoms are $79 + 79 = 158$, $79 + 81 = 160$ and $81 + 81 = 162$. An odd total such as 159 cannot occur.

(b)

• $A_\text{r} = \dfrac{(62.93 \times 69.15) + (64.93 \times 30.85)}{100}$ OR $43.516 + 20.031$ ✓

• $63.55$ ✓

Award [2] for correct final answer. Do not accept the data booklet value without working.

(c)

• 29 protons AND 34 neutrons AND 27 electrons ✓

(d)

• same number of protons/nuclear charge AND same number of electrons ✓

• same electron configuration «so they react in the same way»/neutrons do not affect chemical reactions ✓

Do not accept "same number of neutrons".