S1.3 Electron configurations

The electron configuration of an atom describes how its electrons are arranged in energy levels, sublevels and orbitals. It explains the chemistry of every element: which ions an atom forms, how many bonds it makes and where it sits in the periodic table. Most of the evidence for this arrangement comes from light, such as the yellow-orange glow of a sodium street lamp or the red of a neon sign, each produced by atoms that emit only certain wavelengths.

Light and the electromagnetic spectrum

Visible light is one small part of the electromagnetic spectrum, the whole range of electromagnetic radiation. All electromagnetic radiation travels through a vacuum at the same speed, the speed of light, $c = 3.00\times10^{8}\ \text{m s}^{-1}$. The different regions differ in their wavelength and frequency.

For all electromagnetic waves, wavelength and frequency are inversely proportional: a wave with half the wavelength has twice the frequency. The two are linked through the speed of light by $c = f\lambda$, an equation given in the data booklet.

Photons and photon energy

Electromagnetic radiation is emitted and absorbed in separate packets of energy called photons. The energy of one photon is proportional to the frequency of the radiation. The relationship is written $E = hf$, where $h$ is a constant called the Planck constant.

Putting the two relationships together gives the key chain of reasoning for the whole spectrum:

• shorter wavelength $\Rightarrow$ higher frequency $\Rightarrow$ higher energy per photon;

• longer wavelength $\Rightarrow$ lower frequency $\Rightarrow$ lower energy per photon.

The electromagnetic spectrum is drawn on a logarithmic scale because its wavelengths range over more than twenty powers of ten, from gamma rays shorter than an atomic nucleus to radio waves many metres long.

The regions of the spectrum

The regions blend into each other, so the boundaries are approximate. Each region interacts with matter in a different way, which is why chemists use different regions to probe different features of atoms and molecules.

Wavelength ranges are approximate; the data booklet shows the regions on a logarithmic scale.

Colour, wavelength and energy in visible light

The visible region runs from about 400 nm to 700 nm ($1\ \text{nm} = 10^{-9}\ \text{m}$). Within it, the colours follow the order of the rainbow, violet, blue, green, yellow, orange and red, from short to long wavelength. Violet light therefore has the highest frequency and the most energetic photons in the visible region, and red light the lowest.

Wavelength boundaries from the colour wheel in the IB Chemistry data booklet (2025).

Continuous and line spectra

When light is passed through a prism or a diffraction grating, its different wavelengths are spread out into a spectrum. What the spectrum looks like depends on the source of the light.

White light from a filament lamp or a hot solid gives a continuous spectrum: the colours merge smoothly from violet to red. A gas of one element, heated or excited by an electric discharge, gives a line spectrum instead. Each line has one exact wavelength, and the pattern of lines is different for every element.

Emission and absorption spectra

Line spectra are seen in two complementary forms. An emission spectrum is the light given out by excited atoms: bright coloured lines on a dark background. An absorption spectrum is obtained by shining white light through a cool gas: the gas absorbs certain wavelengths, so the spectrum is continuous apart from dark lines. For a given element the dark absorption lines appear at exactly the same wavelengths as its bright emission lines.

Sunlight looks like a continuous spectrum, but under high magnification it is crossed by hundreds of dark absorption lines. They are produced as atoms in the cooler outer layers of the Sun absorb particular wavelengths of the light coming from the hotter layers beneath.

Observing an emission spectrum

In the laboratory, emission spectra of gaseous elements are produced with a gas discharge tube: a sealed glass tube containing the gas at low pressure, with a high voltage across electrodes at each end. The electric discharge gives the atoms energy, and they emit light. Viewing that light through a hand-held spectroscope (a slit and a diffraction grating, often with a wavelength scale) separates it into lines.

The method gives both qualitative data (the colour and pattern of the lines) and quantitative data (the wavelength of each line, read from the scale). Typical sources of uncertainty are the width of the slit, which broadens each line, the precision of the scale (often about $\pm 5\ \text{nm}$ in a school spectroscope), and faint lines at the violet end, which are hard to see. A spectroscope reading of 650–660 nm for the bright red line of hydrogen, compared with the accepted value of 656.3 nm, is typical.

Metal ions give their characteristic colours in a flame test. A clean nichrome or platinum wire is dipped in concentrated hydrochloric acid and then in the solid, and held at the edge of a hot, non-luminous Bunsen flame. The heat excites electrons in the metal atoms or ions, and the flame takes on the colour of their strongest emission lines.

Fireworks use the same effect: strontium salts give red light, barium salts green and copper salts blue. A flame colour is the combined light of several emission lines, so two elements with similar colours, such as lithium and strontium, are told apart only by passing the light through a spectroscope.

Line spectra as evidence for elements

Because no two elements have the same pattern of lines, a line spectrum identifies an element, much like a fingerprint. Light from a sodium street lamp, for example, consists almost entirely of two lines very close together, at 589.0 nm and 589.6 nm, which is why such lamps look monochromatic yellow-orange. A neon sign glows red-orange because most of the bright lines in neon’s visible spectrum are in the orange and red region.

Spectra can identify elements far beyond the laboratory. During a solar eclipse in 1868, astronomers recorded a bright yellow line at about 588 nm in the light from the Sun’s outer atmosphere that matched no element then known. The new element was named helium, after the Greek helios, the Sun, and it was not found on Earth until 1895. Astronomers still use absorption and emission lines to measure the composition of stars and distant galaxies.

The hydrogen emission spectrum and energy levels

Hydrogen has the simplest atom, with a single electron, and the simplest spectrum. Explaining it was the first step to the modern model of the atom.

How an emission line is produced

The electron in a hydrogen atom can have only certain fixed amounts of energy, called energy levels. Normally it occupies the lowest level, $n = 1$: the atom is in its ground state. When the atom absorbs energy, from heat or an electric discharge, the electron moves up to a higher level and the atom is in an excited state.

An excited state is unstable. The electron soon falls back to a lower level, and the energy it loses is emitted as a single photon. The photon’s energy is exactly the difference in energy between the two levels:

The explanation of a line spectrum follows in three linked steps:

1. the electron can occupy only certain energy levels, so only certain energy differences $\Delta E$ are possible;

2. each fall between two particular levels emits a photon of one particular energy, and therefore one particular frequency and wavelength;

3. each possible fall gives one line in the spectrum, at a fixed position.

If the electron could have any energy at all, any energy difference would be possible and the atom would emit every wavelength, giving a continuous spectrum. The observed line spectrum is therefore evidence that the energy of electrons in atoms is quantized.

The series of lines in the hydrogen spectrum

Hydrogen’s lines fall into groups (series), each made up of all the transitions that end on the same level. The energy-level diagram of hydrogen shows the three series that matter for chemistry.

• Falls to $n = 1$ involve the largest energy gaps, so their photons have the highest energies. These lines are in the ultraviolet region.

• Falls to $n = 2$ give the four lines in the visible region, together with further lines just into the ultraviolet.

• Falls to $n = 3$ involve the smallest gaps of the three series and give lines in the infrared region.

The three series were historically named after the scientists who studied them (Lyman, Balmer and Paschen); what matters chemically is the level on which each series ends.

The visible series is the one seen through a spectroscope in a school laboratory. Its four bright lines are red (656 nm), blue-green (486 nm) and two violet lines (434 nm and 410 nm).

Each visible line comes from one particular fall to $n = 2$. The bigger the fall, the more energetic the photon and the shorter its wavelength. The red line, the lowest-energy line in the visible series, comes from the smallest fall, $n = 3 \to n = 2$.

Why the lines converge

The energy levels of the hydrogen atom are not evenly spaced. The gap between $n = 1$ and $n = 2$ is very large; the gap between $n = 2$ and $n = 3$ is much smaller, and the gaps keep shrinking as $n$ increases. At very high $n$ the levels crowd together, approaching a limit labelled $n = \infty$, at which the electron is no longer held by the nucleus.

Because of this, the lines in each series get closer together towards the high-frequency (short-wavelength) end of the series. The red and blue-green lines of the visible series are far apart, but the violet lines crowd together, and beyond them the lines become too close to separate. This merging of lines is called convergence.

Comparing transitions

The energy of a transition depends only on the gap between the two levels, which depends on both levels, not just on how many levels the electron passes. Because the levels converge, a fall between two high levels releases much less energy than a fall between two low ones. The fall $n = 4 \to n = 3$ releases less energy than $n = 3 \to n = 2$, which in turn releases far less than $n = 2 \to n = 1$.

The energy-level model and its limitations

The picture of an electron moving between fixed energy levels comes from the model of the atom proposed by Niels Bohr in 1913. It explains the hydrogen spectrum very well: the energies of the levels it predicts match the measured wavelengths of the lines. Like every scientific model, it has limitations:

• it treats the electron as a particle moving in a fixed circular path around the nucleus, whereas the position of an electron can only be described by a probability;

• it works accurately only for atoms or ions with one electron, such as H and He⁺, and cannot predict the spectra of atoms with more electrons;

• it does not show the sublevels (s, p, d and f) and orbitals within each main level, or explain why each level holds a particular number of electrons;

• it gives no reason why only certain energy levels are allowed.

The modern quantum mechanical model keeps the idea of quantized energy levels but replaces fixed paths with orbitals, regions where an electron is likely to be found.

Main energy levels and how many electrons they hold

The electrons in any atom occupy main energy levels (often called shells), numbered $n = 1, 2, 3, 4, \ldots$ outwards from the nucleus. The number $n$ is the principal quantum number. Electrons in higher main levels are, on average, further from the nucleus and higher in energy.

The rule $2n^2$ follows from the sublevels and orbitals inside each main level: level $n$ contains $n^2$ orbitals, and each orbital holds two electrons.

The number of the highest main level that contains electrons equals the period of the element in the periodic table. Sodium, in period 3, has electrons in levels 1, 2 and 3; potassium, in period 4, has its outermost electron in level 4.

Sublevels and atomic orbitals

The line spectra of atoms with more than one electron contain more lines than the simple energy-level model can explain. The extra detail comes from a more refined model, in which each main level is divided into sublevels of slightly different energy.

Sublevels

The first main level contains only an s sublevel (1s). The second contains 2s and 2p, the third 3s, 3p and 3d, and the fourth 4s, 4p, 4d and 4f. Each type of sublevel always contains the same number of orbitals.

Adding up the sublevels gives the capacity of each main level. Level 3, for example, holds $2 + 6 + 10 = 18$ electrons in $1 + 3 + 5 = 9$ orbitals, and level 4 holds $2 + 6 + 10 + 14 = 32$ electrons in 16 orbitals.

The relative energies of the sublevels decide the order in which they fill. The energy diagram of the sublevels up to 4p shows that the main levels begin to overlap from $n = 3$ onwards: the 4s sublevel is slightly lower in energy than 3d.

Atomic orbitals

The exact position and path of an electron cannot be known. Instead, the quantum mechanical model gives the probability of finding an electron in each region around the nucleus.

An atomic orbital is usually drawn as a boundary surface that encloses the region where the electron is found most of the time (for example, 90 % of the time). Its shape depends on the type of sublevel, and its size grows with $n$.

s orbitals

Every s orbital is spherical and centred on the nucleus, so the probability of finding the electron depends only on the distance from the nucleus, not on the direction. The 2s orbital has the same shape as 1s but is larger, because a 2s electron is on average further from the nucleus.

p orbitals

A p orbital has two lobes, one on each side of the nucleus, giving a dumbbell shape. Each p sublevel has three p orbitals of equal energy. They lie along three mutually perpendicular axes and are labelled $\text{p}_x$, $\text{p}_y$ and $\text{p}_z$. A 3p orbital has the same shape as a 2p orbital but is larger.

The five d orbitals and seven f orbitals have more complex shapes. Their energies matter for writing configurations, but their shapes are not needed.

Electron spin and the Pauli exclusion principle

Electrons have a property called spin, which can take one of two values. In orbital diagrams the two spin states are drawn as an arrow pointing up (↑) and an arrow pointing down (↓). Two electrons in the same orbital must have opposite spins.

The Pauli exclusion principle sets the limit of two electrons per orbital, and so produces all the capacities in the sublevel table: 2 for s, 6 for p, 10 for d and 14 for f. Two electrons occupying one orbital with opposite spins are described as a spin pair, or simply a pair; a single electron in an orbital is an unpaired electron.

Writing electron configurations of atoms

The ground-state electron configuration of an atom is built up by adding electrons one at a time to the available orbitals. Three rules decide where each electron goes.

The Aufbau principle and the order of filling

Following the Aufbau principle for the first 36 elements gives this order of filling:

The one step that does not follow the numbering is 4s before 3d. For potassium and calcium the 4s orbital is lower in energy than the 3d orbitals, so the 19th and 20th electrons enter 4s. Only from scandium onwards do electrons enter 3d. A simple way to recall the whole order is to write the sublevels in rows and read along the diagonals.

Hund’s rule

Electrons repel each other. Two electrons in separate orbitals of the same sublevel are further apart, and repel each other less, than two electrons in one orbital, so the arrangement with unpaired electrons in separate orbitals is lower in energy. Hund's rule matters in the p, d and f sublevels, which have more than one orbital of the same energy.

Orbital diagrams

An orbital diagram (arrow-in-box diagram) shows every orbital as a box and every electron as an arrow, so it shows the filling, the spins and the relative energies of the orbitals at a glance. The three orbitals of a p sublevel, or the five of a d sublevel, are drawn as boxes joined in a row because they have the same energy. The orbital diagrams of carbon, nitrogen and oxygen show all three rules at work.

Full and condensed electron configurations

An electron configuration is written as a list of sublevels, each with a superscript giving the number of electrons in it. Sodium (11 electrons) has the full configuration $\text{1s}^{2}\ \text{2s}^{2}\ \text{2p}^{6}\ \text{3s}^{1}$.

Because the inner electrons of many elements match a noble gas configuration, a shorter condensed form is often used. The noble gas with the configuration of the inner electrons is written in square brackets, followed by the remaining electrons. Sodium becomes $[\text{Ne}]\ \text{3s}^{1}$, because the configuration of neon is $\text{1s}^{2}\ \text{2s}^{2}\ \text{2p}^{6}$.

From period 4 onwards, configurations can be written either in the order of filling (4s before 3d) or grouped by main level (3d before 4s). Iron, for example, is $[\text{Ar}]\ \text{4s}^{2}\ \text{3d}^{6}$ or $[\text{Ar}]\ \text{3d}^{6}\ \text{4s}^{2}$; both describe the same arrangement and both are accepted. Grouping by main level, as here, makes the configurations of transition element ions easier to write.

Orbital diagrams and configurations together make it easy to count electrons of a particular kind, a skill often tested with phosphorus and the transition elements.

Configurations and the periodic table

The periodic table is built from electron configurations. The period number gives the highest main level occupied, and the block (s, p, d or f) is the sublevel that receives the highest-energy electron.

• s block (groups 1 and 2, plus helium): the outer electrons are in an s sublevel, $n\text{s}^{1}$ or $n\text{s}^{2}$.

• p block (groups 13 to 18): the outer electrons fill a p sublevel, from $n\text{s}^{2}\ n\text{p}^{1}$ in group 13 to $n\text{s}^{2}\ n\text{p}^{6}$ in group 18.

• d block (groups 3 to 12): the 3d sublevel fills across period 4, from scandium to zinc.

• f block (lanthanoids and actinoids): the f sublevels fill, from period 6.

The electrons in the highest occupied main level are the valence electrons of a main-group element. Their number equals the group number for groups 1 and 2, and the group number minus 10 for groups 13 to 18: chlorine (group 17), $[\text{Ne}]\ \text{3s}^{2}\ \text{3p}^{5}$, has seven. Elements in the same group have the same arrangement of valence electrons, which is why they have similar chemical properties. The noble gases have full outer s and p sublevels ($n\text{s}^{2}\ n\text{p}^{6}$, or $\text{1s}^{2}$ for helium), and they are very unreactive.

The exceptions: chromium and copper

Two elements up to $Z = 36$ do not follow the Aufbau order exactly. Chromium is $[\text{Ar}]\ \text{3d}^{5}\ \text{4s}^{1}$, not $[\text{Ar}]\ \text{3d}^{4}\ \text{4s}^{2}$, and copper is $[\text{Ar}]\ \text{3d}^{10}\ \text{4s}^{1}$, not $[\text{Ar}]\ \text{3d}^{9}\ \text{4s}^{2}$. In each, one electron that would be expected in 4s is in 3d instead.

The 3d and 4s sublevels are very close in energy. In chromium, moving one electron from 4s to 3d gives six unpaired electrons in a half-filled 3d sublevel, with no electrons paired in the same orbital, and this arrangement has a lower total energy because there is less repulsion between paired electrons. In copper, the move gives a completely filled 3d sublevel, which again turns out to be the lower-energy arrangement. In both cases the electron density of the 3d sublevel is spread evenly around the nucleus.

Electron configurations of ions

An ion has gained or lost electrons, so its configuration is found by starting from the configuration of the atom and then adding or removing electrons.

Negative ions

A negative ion (anion) has extra electrons, which go into the next available orbitals in the usual order. A sulfur atom, $[\text{Ne}]\ \text{3s}^{2}\ \text{3p}^{4}$, gains two electrons to form the sulfide ion, ${{\mathrm{S}{\vphantom{X}}^{2-}}}$: $[\text{Ne}]\ \text{3s}^{2}\ \text{3p}^{6}$, which is the same as $[\text{Ar}]$. Chlorine, $[\text{Ne}]\ \text{3s}^{2}\ \text{3p}^{5}$, gains one electron to form ${{\mathrm{Cl}{\vphantom{X}}^{-}}}$, also $[\text{Ne}]\ \text{3s}^{2}\ \text{3p}^{6}$.

Positive ions of main-group elements

A positive ion (cation) of a main-group element is formed by removing electrons from the highest occupied main level, the outermost electrons. Magnesium, $[\text{Ne}]\ \text{3s}^{2}$, loses its two 3s electrons to form ${{\mathrm{Mg}{\vphantom{X}}^{2+}}}$, $\text{1s}^{2}\ \text{2s}^{2}\ \text{2p}^{6}$; aluminium, $[\text{Ne}]\ \text{3s}^{2}\ \text{3p}^{1}$, loses three to form ${{\mathrm{Al}{\vphantom{X}}^{3+}}}$, also $\text{1s}^{2}\ \text{2s}^{2}\ \text{2p}^{6}$.

Isoelectronic species

Many simple ions have the configuration of a noble gas. Species with the same electron configuration are described as isoelectronic.

Isoelectronic species differ in their nuclear charge, the total positive charge of the nucleus, equal to the number of protons. With the same number of electrons, the species with more protons attracts its electrons more strongly and is smaller. In the series ${{\mathrm{S}{\vphantom{X}}^{2-}}}$, ${{\mathrm{Cl}{\vphantom{X}}^{-}}}$, ${{\mathrm{Ar}}}$, ${{\mathrm{K}{\vphantom{X}}^{+}}}$, ${{\mathrm{Ca}{\vphantom{X}}^{2+}}}$ (all $[\text{Ne}]\ \text{3s}^{2}\ \text{3p}^{6}$), the nuclear charge rises from 16+ to 20+, so the radius decreases from ${{\mathrm{S}{\vphantom{X}}^{2-}}}$ to ${{\mathrm{Ca}{\vphantom{X}}^{2+}}}$. The radii of ions are developed further with the ionic model and periodic trends.

Positive ions of transition elements

The transition elements of period 4 follow one extra rule: when they form positive ions, the 4s electrons are removed before the 3d electrons. Iron, $[\text{Ar}]\ \text{3d}^{6}\ \text{4s}^{2}$, loses its two 4s electrons to form ${{\mathrm{Fe}{\vphantom{X}}^{2+}}}$, $[\text{Ar}]\ \text{3d}^{6}$. Losing one more electron, from 3d, gives ${{\mathrm{Fe}{\vphantom{X}}^{3+}}}$, $[\text{Ar}]\ \text{3d}^{5}$.

The 4s sublevel fills before 3d, yet it empties first. The explanation is that the relative energies of the sublevels change once the 3d orbitals contain electrons. In the atoms and ions of transition elements, the 3d electrons are on average closer to the nucleus than the 4s electrons, and 3d becomes lower in energy than 4s. The 4s electrons are then the outermost, highest-energy electrons, so they are the first to be removed.

The ability of transition elements to lose different numbers of 3d electrons, after the 4s electrons, is the basis of their variable oxidation states, studied with the properties of transition elements.

Photon energy and the convergence limit (HL)

Calculating the energy of a photon

The energy $E$ of one photon is proportional to its frequency $f$, with the Planck constant, $h$, as the constant of proportionality. Frequency and wavelength $\lambda$ are linked by the speed of light $c$. Both equations, and both constants, are in the data booklet:

Here $h = 6.63\times10^{-34}\ \text{J s}$ and $c = 3.00\times10^{8}\ \text{m s}^{-1}$. Combining them gives $E = hc/\lambda$, so the energy gap behind any spectral line can be found from its frequency or its wavelength.

The same method gives the energy gap between two orbitals in any atom. The yellow light of a sodium street lamp, wavelength $589\ \text{nm}$, comes from electrons falling from the 3p to the 3s orbital of sodium. Each photon carries $E = hc/\lambda = (6.63\times10^{-34})(3.00\times10^{8})/(5.89\times10^{-7}) = 3.38\times10^{-19}\ \text{J}$, which is $203\ \text{kJ mol}^{-1}$ for a mole of sodium atoms.

The convergence limit and ionization

In each series of the hydrogen spectrum, the lines crowd together and finally merge at a frequency called the convergence limit. The line at the convergence limit corresponds to an electron falling from $n = \infty$, the edge of the atom, to the level on which the series ends. For the ultraviolet series, which ends on $n = 1$, this energy is the same as the energy needed to move an electron from $n = 1$ to $n = \infty$: to remove it completely from a hydrogen atom in its ground state.

Removing an electron completely from an atom is called ionization. On an energy-level diagram it is drawn as an upward arrow from the ground state, $n = 1$, to $n = \infty$.

First ionization energy

The first ionization energy is measured in $\text{kJ mol}^{-1}$ and is always positive (endothermic), because energy must be supplied to overcome the attraction between the electron and the nucleus. The atoms must be gaseous so that the value refers to isolated atoms, without attractions to neighbouring atoms. The data booklet lists first ionization energies for the elements.

The convergence limit of the hydrogen spectrum therefore gives a direct measurement of the first ionization energy of hydrogen: find the energy of one photon at the convergence limit, then multiply by the Avogadro constant to obtain the value per mole.

When the data are given as a wavelength, the photon energy is found from $E = hc/\lambda$ instead. The convergence limit of the same series lies at $91.2\ \text{nm}$, and $(6.63\times10^{-34})(3.00\times10^{8})/(91.2\times10^{-9}) = 2.18\times10^{-18}\ \text{J}$ gives the same result.

Trends in first ionization energy (HL)

The three factors

The first ionization energy measures how strongly an atom holds its outermost electron. Three factors decide the size of that attraction:

• Nuclear charge. More protons in the nucleus attract the electron more strongly.

• Distance from the nucleus. An electron in a higher main level is, on average, further from the nucleus and is attracted less strongly. Within a main level, a p electron is slightly higher in energy than an s electron.

• Shielding. Electrons in inner main levels repel the outer electron and partly cancel the attraction of the nucleus.

Electrons in the same main level shield each other only slightly, while electrons in inner levels shield the outer electrons strongly. The outer electron of sodium, for example, is held by the 11+ nucleus but shielded by ten inner electrons, so the net attraction it feels is far smaller than that of the full 11+ charge.

A graph of first ionization energy against atomic number, compiled from a database of values such as the data booklet, shows these factors at work as a repeating pattern across the periods.

Across a period

First ionization energy generally increases across a period, from a minimum at the group 1 metal to a maximum at the noble gas. Across a period:

• the nuclear charge increases by one with each element;

• the electrons are added to the same main level, so the shielding by inner electrons stays about the same;

• the outer electrons are therefore attracted more strongly and the atomic radius decreases, so more energy is needed to remove one.

Down a group

First ionization energy decreases down a group, for example from lithium (520) to sodium (496) to potassium ($419\ \text{kJ mol}^{-1}$). Going down the group:

• the outer electron is in a higher main level, further from the nucleus;

• there are more inner electrons shielding it;

• these effects outweigh the increase in nuclear charge, so the outer electron is held less strongly and is easier to remove.

The large drop from each noble gas to the next alkali metal has the same cause: the alkali metal’s outer electron starts a new main level, further out and well shielded.

These trends underlie the division of the elements into metals and non-metals. Elements with low first ionization energies, towards the left of each period and the bottom of each group, lose their outer electrons readily and form positive ions: they show metallic behaviour. Elements with high first ionization energies, towards the top right of the periodic table, hold their electrons strongly and tend instead to gain or share electrons, as non-metals do.

Discontinuities across a period: evidence for sublevels

The general rise across periods 2 and 3 is interrupted by two small drops: from group 2 to group 13 (Be → B and Mg → Al) and from group 15 to group 16 (N → O and P → S). These dips cannot be explained by main levels alone; they are evidence for sublevels.

From group 2 to group 13

Beryllium is $\text{1s}^{2}\ \text{2s}^{2}$ and boron $\text{1s}^{2}\ \text{2s}^{2}\ \text{2p}^{1}$. The electron removed from boron is a 2p electron, which is higher in energy than the 2s electron removed from beryllium. It is also slightly further from the nucleus and is partly shielded by the 2s electrons. Less energy is needed to remove it, even though boron has the larger nuclear charge: $801\ \text{kJ mol}^{-1}$ compared with $900\ \text{kJ mol}^{-1}$. The same reasoning explains why aluminium (578, removing a 3p electron) is below magnesium (738, removing a 3s electron).

From group 15 to group 16

Nitrogen is $\text{1s}^{2}\ \text{2s}^{2}\ \text{2p}^{3}$, with one electron in each 2p orbital. Oxygen is $\text{1s}^{2}\ \text{2s}^{2}\ \text{2p}^{4}$, so one of its 2p orbitals holds a pair of electrons. The two electrons in that orbital repel each other strongly, which raises the energy of each, and one of them is removed more easily than an unpaired electron. Oxygen’s first ionization energy (1314) is therefore lower than nitrogen’s (1402) despite its greater nuclear charge. Phosphorus (1012) and sulfur ($1000\ \text{kJ mol}^{-1}$) show the same pattern.

Nitrogen is higher than both its neighbours, for two different reasons. It is higher than carbon because it has one more proton with similar shielding (the general trend); it is higher than oxygen because oxygen loses one of a pair of electrons that repel each other.

Why ionization energies are plotted on log scales

Ionization energies range from about $400\ \text{kJ mol}^{-1}$ for potassium to more than $200\,000\ \text{kJ mol}^{-1}$ for the last electrons of aluminium. On a linear scale the small values would be squashed together near the axis. A logarithmic scale plots $\log_{10}$ of the value, so each step of 1 on the axis is a factor of ten; values spanning several orders of magnitude fit on one graph, and the large jumps and small changes are both visible. Chemists use the same idea for hydrogen ion concentrations, which the pH scale compresses into numbers from about 0 to 14.

Successive ionization energies (HL)

Once the first electron has gone, further electrons can be removed one at a time. The energies needed are the second, third and later successive ionization energies. Each refers to gaseous ions losing one more electron:

Why successive ionization energies always increase

Each successive ionization energy is larger than the one before. After each electron is removed, the remaining electrons are held by the same nuclear charge but repel each other less, and the ion is more positive, so the electrons are pulled closer to the nucleus. Each further electron is held more strongly and needs more energy to remove.

Big jumps show the main levels

The increases are not even. A large jump occurs whenever the next electron has to come from a main level closer to the nucleus. That electron is much nearer the nucleus and much less shielded, because only the electrons in levels further in now shield it, so it needs far more energy to remove.

The successive ionization energies of aluminium, plotted on a log scale, show this clearly.

• Electrons 1 to 3 are removed from the third main level ($\text{3p}^1$, then $\text{3s}^2$). Their energies are low: 578, 1817 and $2745\ \text{kJ mol}^{-1}$.

• The fourth electron comes from the second main level (2p), closer to the nucleus and shielded only by the two 1s electrons. Its energy, $11\,577\ \text{kJ mol}^{-1}$, is about four times the third: the first big jump.

• Electrons 4 to 11 come from the second main level and increase steadily.

• Electrons 12 and 13 come from the first main level, right next to the nucleus and not shielded at all, giving the second big jump (about $201\,000\ \text{kJ mol}^{-1}$).

The pattern of jumps, three electrons, then eight, then two, matches the configuration $\text{1s}^{2}\ \text{2s}^{2}\ \text{2p}^{6}\ \text{3s}^{2}\ \text{3p}^{1}$ read from the outside in. Successive ionization energies are some of the strongest evidence for the existence of main energy levels. Smaller breaks in the data also show the sublevels: for aluminium, the gap between the first ionization energy (the single 3p electron) and the second (from 3s) is larger than the gaps within the 3s pair.

Deducing the group of an element

The number of electrons removed before the first big jump equals the number of electrons in the outer main level. For main-group elements this identifies the group: one electron before the jump means group 1, two means group 2, three means group 13, and so on up to eight for group 18 (group number minus 10 for the p block).

Energy needed to form an ion

To form an ion with a charge of 2+ or more, all the successive ionizations up to that charge must happen, so their energies are added. The successive ionization energies of magnesium, in $\text{kJ mol}^{-1}$, start 738, 1451, 7733, 10543, …; the energy needed to form one mole of ${{\mathrm{Mg}{\vphantom{X}}^{2+}\mskip2mu (\mathrm{g})}}$ from ${{\mathrm{Mg}\mskip2mu (\mathrm{g})}}$ is the sum of the first two:

The third ionization energy of magnesium, $7733\ \text{kJ mol}^{-1}$, is more than five times the second, because the third electron would come from the second main level. Magnesium therefore forms ${{\mathrm{Mg}{\vphantom{X}}^{2+}}}$ and never ${{\mathrm{Mg}{\vphantom{X}}^{3+}}}$ in its compounds, because the energy released in forming bonds or a lattice cannot repay such a large ionization energy.

Comparing ionization energies of different species

The same three factors explain comparisons between ions and atoms. The second ionization energy of aluminium ($1817\ \text{kJ mol}^{-1}$) is much larger than the first ionization energy of magnesium ($738\ \text{kJ mol}^{-1}$), although both remove a 3s electron from a species with the configuration $[\text{Ne}]\ \text{3s}^{2}$. The aluminium ion ${{\mathrm{Al}{\vphantom{X}}^{+}}}$ has one more proton (13 rather than 12) acting on the same number of electrons, and it is already positively charged, so its electrons are held more strongly.

Successive ionization energies also show the sulfur atom’s configuration, $[\text{Ne}]\ \text{3s}^{2}\ \text{3p}^{4}$: the first six values rise fairly steadily, with a slightly larger step between the fourth (the last 3p electron) and the fifth (the first 3s electron), and then the seventh is about three times the sixth, because it comes from the second main level. In iron, the large rise between the eighth and ninth ionization energies marks the point where all eight 4s and 3d electrons have gone and the next electron comes from the 3p sublevel, which is lower in energy and closer to the nucleus than 3d and is not shielded by the 3d electrons. The patterns in the successive ionization energies of the transition elements also help to explain their variable oxidation states.

Applying the ideas in exam questions

Explanations of ionization energy differences follow the same pattern as the trends: name the change in nuclear charge, then the change in distance, shielding or electron pairing, giving one marking point each.

Paper 1A: the configuration of a copper ion

Paper 1A: the lowest-energy transition

Paper 1A: the group from successive ionization energies (HL)

Paper 1B: measuring the hydrogen spectrum

Paper 2: electron configurations of manganese

Paper 2: sodium, spectra and ionization energy (HL)

Self-check

• To do:

  I can describe the relationship between colour, wavelength, frequency and photon energy across the electromagnetic spectrum.

• To do:

  I can distinguish between a continuous spectrum and a line spectrum, and between emission and absorption spectra.

• To do:

  I can explain how the lines of an emission spectrum are produced and why they are evidence for discrete energy levels.

• To do:

  I can describe the hydrogen emission spectrum, relating the ultraviolet, visible and infrared lines to falls to n = 1, 2 and 3, and explain why the lines converge.

• To do:

  I can identify and compare transitions on an energy-level diagram, drawing emission arrows down and absorption arrows up.

• To do:

  I can state limitations of the energy-level (Bohr) model of the atom.

• To do:

  I can deduce the maximum number of electrons in a main energy level (2n²), a sublevel and an orbital.

• To do:

  I can name the sublevels in each main level and state their relative energies.

• To do:

  I can recognize and sketch the shapes of s orbitals and the three p orbitals.

• To do:

  I can state and apply the Aufbau principle, Hund’s rule and the Pauli exclusion principle.

• To do:

  I can write full and condensed electron configurations and draw orbital diagrams for atoms from H to Kr, including Cr and Cu.

• To do:

  I can relate electron configurations to the period, group and block of an element.

• To do:

  I can write the configurations of ions, including transition element ions, and identify isoelectronic species.

• To do:

  I can calculate the energy of a photon from its frequency or wavelength (HL).

• To do:

  I can explain how the convergence limit relates to ionization and calculate the first ionization energy from it (HL).

• To do:

  I can explain the trends and discontinuities in first ionization energy across a period and down a group (HL).

• To do:

  I can explain the pattern of successive ionization energies, deduce the group of an element from them, and calculate the energy to form an ion (HL).

The answer is A.

And 13 more in the data booklet.

(c) $n = 1 \to 3$ is an absorption (the electron moves up), so it does not produce an emission line. $n = 3 \to 1$ is an emission ending on $n = 1$, which is in the ultraviolet. $n = 4 \to 3$ is an emission ending on $n = 3$, so it is the infrared line.

(b) ${{\mathrm{Br}}}$ is $[\text{Ar}]\ \text{3d}^{10}\ \text{4s}^{2}\ \text{4p}^{5}$; adding one electron to 4p gives ${{\mathrm{Br}{\vphantom{X}}^{-}}}$: $[\text{Ar}]\ \text{3d}^{10}\ \text{4s}^{2}\ \text{4p}^{6}$ (that is, $[\text{Kr}]$).

(c) ${{\mathrm{Cr}}}$ is $[\text{Ar}]\ \text{3d}^{5}\ \text{4s}^{1}$ (an exception). Remove the 4s electron, then two 3d electrons: ${{\mathrm{Cr}{\vphantom{X}}^{3+}}}$ is $[\text{Ar}]\ \text{3d}^{3}$.

(d) ${{\mathrm{Cu}}}$ is $[\text{Ar}]\ \text{3d}^{10}\ \text{4s}^{1}$ (an exception). Remove the 4s electron, then one 3d electron: ${{\mathrm{Cu}{\vphantom{X}}^{2+}}}$ is $[\text{Ar}]\ \text{3d}^{9}$.

(e) ${{\mathrm{Ca}{\vphantom{X}}^{2+}}}$ has 18 electrons, the same configuration as argon.

Energy for one mole of atoms, using $N_\text{A} = 6.02\times10^{23}\ \text{mol}^{-1}$:

Convert to kilojoules: $1.31\times10^{6}\ \text{J mol}^{-1} \div 1000 = 1.31\times10^{3}\ \text{kJ mol}^{-1}$ (1313 to four significant figures before rounding).

The data booklet value is $1312\ \text{kJ mol}^{-1}$, so the answer agrees to three significant figures.

(b)

• 5 ✓

(c)

• ${{\mathrm{Mn}{\vphantom{X}}^{4+}}}$: $[\text{Ar}]\ \text{3d}^{3}$ ✓

• Do not accept $[\text{Ar}]\ \text{3d}^{1}\ \text{4s}^{2}$.

(d)

• five 3d boxes, each with one arrow, all parallel AND 4s box empty or not shown ✓

• Accept all arrows pointing down. Accept half-arrows.

(e)

• maximum 18 AND 13 present «$\text{3s}^2\ \text{3p}^6\ \text{3d}^5$» ✓

(f)

• $[\text{Ar}]\ \text{3d}^{5}\ \text{4s}^{1}$ ✓

• Accept $[\text{Ar}]\ \text{4s}^{1}\ \text{3d}^{5}$.

(b)

• «minimum» energy required to remove one mole of electrons ✓

• from one mole of gaseous atoms «in their ground state» ✓

• Accept the equation ${{\mathrm{Na}\mskip2mu (\mathrm{g})\xrightarrow{}\mathrm{Na}{\vphantom{X}}^{+}\mskip2mu (\mathrm{g}) {}+{} \mathrm{e}{\vphantom{X}}^{-}}}$ for M2.

(c)

• magnesium has a greater nuclear charge/more protons ✓

• outer electrons of both are in the same main level/3s, so shielding is similar/magnesium has a smaller atomic radius ✓

• Do not accept "magnesium has a full sublevel" or "more electrons".

(d)

• the second electron is removed from a main level closer to the nucleus/n = 2 rather than n = 3 ✓

• which is less shielded, so the electron is held «much» more strongly ✓

• Accept "removed from a positive ion" as a supporting point, but not for either mark on its own.

(e)

• sulfur $\text{3p}^{4}$ has a pair of electrons in one 3p orbital, while the 3p electrons of phosphorus $\text{3p}^{3}$ are unpaired ✓

• repulsion between the paired electrons makes one of them easier to remove ✓

• Accept orbital diagrams for M1. Do not accept "half-filled sublevel of phosphorus is more stable".