Light as evidence for electron energy changes

An electromagnetic spectrum is the range of electromagnetic radiation arranged by wavelength or frequency. Visible light makes up only a small section of it. Red light has a longer wavelength and lower frequency than violet light. As you move from radio waves through microwaves, infrared, visible, ultraviolet, X-rays and gamma rays, wavelength decreases, while frequency and photon energy increase.

The relationship is c = λf, where c is the speed of light in a vacuum (m s⁻¹), λ is wavelength (m) and f is frequency (s⁻¹). Photon energy is given by E = hf, where E is the energy of one photon (J) and h is the Planck constant (J s). Higher frequency, then, means higher energy; longer wavelength means lower energy. Keep the colour–energy link in mind: red is lower energy than blue or violet.

A photon is a packet of electromagnetic energy absorbed or emitted as a single unit. In atoms, electrons lose energy by emitting photons. An excited state is an energy state of an atom in which at least one electron has more energy than it has in the lowest-energy arrangement. When an electron in an excited state falls back to a lower energy level, the atom emits a photon with energy equal to the energy difference between the two levels.

Continuous spectra and line spectra

A continuous spectrum contains an unbroken range of wavelengths, with no gaps between colours. White light split by a prism gives this type of spectrum.

A line spectrum contains only particular wavelengths, seen as separate bright or dark lines. A hot, low-pressure gaseous element gives a line emission spectrum: a line spectrum of bright lines on a dark background, because atoms emit photons at specific wavelengths. A cold gas in front of continuous light can give an absorption spectrum, with dark lines at the wavelengths that have been absorbed.

The difference is useful because a line emission spectrum acts as a fingerprint for an element. Different elements have different allowed electron energy changes, so they emit different sets of wavelengths. A spectroscope therefore shows more than our eyes can. Two lamps may look similar in colour, but their line spectra can be quite different.

In a simple practical investigation, a gas discharge tube supplies excited gaseous atoms, and a prism or diffraction grating separates the emitted light. Qualitative data include the colours and number of visible lines; quantitative data include measured wavelengths. Good records compare observed wavelengths with reference values, often by plotting observed wavelength against accepted wavelength and judging the accuracy from the agreement with a best-fit line.

Why hydrogen gives lines, not a smear

Hydrogen is the simplest place to start, because it has just one electron. Its line emission spectrum shows something crucial: the electron cannot take any energy value it wants. If every energy were allowed, the spectrum would be continuous. Instead, the electron sits in discrete energy levels, which are allowed electron energy states separated by gaps in energy.

A ground state is the lowest-energy state available to an electron in an atom. In hydrogen, the electron is in the ground state when it is at the first energy level. If it absorbs exactly the right amount of energy, it can jump to a higher level. When it drops back down, it emits a photon. The photon energy equals the energy gap between the two levels, so each downward transition produces a particular wavelength.

Hydrogen’s energy levels crowd closer together at higher energies. That is why the spectral lines bunch together, or converge, towards the high-frequency end of a series. Do not learn the historical series names for IB; learn what the transitions mean.

Transitions to the first, second and third energy levels

Transitions that end at the first energy level release relatively large amounts of energy, so their photons are in the ultraviolet region. Transitions ending at the second energy level include the visible hydrogen lines. The familiar red visible line comes from a smaller drop than the blue-violet lines, so it has a longer wavelength, lower frequency and lower energy. Transitions ending at the third energy level release less energy overall and lie in the infrared region.

There’s a useful nature-of-science point here too. Emission spectra provide evidence for different elements because each element has a different nuclear charge and electron arrangement, which gives a different pattern of allowed energy gaps. In astronomy, scientists can match dark absorption lines in starlight with known laboratory lines to infer which elements are present in stars.

When you study emission spectra using gas discharge tubes and prisms, the useful qualitative data are the colours, brightness and order of lines. The useful quantitative data are the wavelengths or frequencies of the lines. The instrument extends our senses: the naked eye may report “pink” or “blue-white”, but the spectrum gives a much more precise pattern.

The principal quantum number

A main energy level is a group of electron states in an atom, identified by a principal quantum number and lying at a broadly similar distance from the nucleus. The principal quantum number, n, is a positive integer that labels a main energy level (unitless). So the first level is n = 1, the second is n = 2, and so on.

A main energy level can hold a maximum of 2n² electrons, where n is the principal quantum number. This gives:

Use the formula with care: it gives the capacity of the whole main energy level, not necessarily the number of electrons actually present in a ground-state atom.

Main energy level capacities calculated using 2n² for n = 1 to 4.

An atom’s highest occupied main energy level links to its period in the periodic table. For main-group elements, the period number gives the highest occupied value of n in the ground-state atom. For example, elements in period 3 have their outer electrons in the third main energy level.

From orbits to orbitals

The Bohr model helps introduce discrete energy levels, but it only goes so far. Electrons are not tiny planets moving around the nucleus in fixed circular paths. In the quantum mechanical model, we talk about where an electron is likely to be found, rather than the exact path it takes.

An atomic orbital is a region of space around a nucleus in which there is a high probability of finding an electron. The words “high probability” matter here. An orbital is not a shell, a track, or a container with a hard wall; it is a mathematical probability region, usually drawn as a boundary surface because that makes it easier to picture.

A sublevel is a subdivision of a main energy level containing orbitals of a particular type. Within a given main level, the sublevels s, p, d and f have successively higher energy. The number of sublevel types available in a main level equals n: for n = 1 only s is available; for n = 2, s and p; for n = 3, s, p and d; for n = 4, s, p, d and f.

Shapes you must recognize

An s orbital is an atomic orbital with spherical symmetry around the nucleus. Higher s orbitals are larger on average, but for IB shape recognition the key point is simple: s is spherical.

A p orbital is an atomic orbital with two lobes on opposite sides of the nucleus, often described as dumbbell-shaped. A p sublevel contains three p orbitals, oriented along the x, y and z axes. These are commonly labelled pₓ, pᵧ and p_z. The shape is the same in each case; only the direction changes.

Periodic table blocks link directly to sublevels. The s-block elements are filling s sublevels, the p-block elements are filling p sublevels, the d-block elements are filling d sublevels, and the f-block elements are filling f sublevels. That structure is why the periodic table is more than a list: its shape reflects electron arrangements.

Orbitals, sublevels and spin

An electron configuration is a notation that shows how the electrons in an atom or ion are arranged in orbitals and sublevels. It matters because the outer, or valence, electrons strongly influence chemical properties.

For a given electron configuration and chemical environment, each orbital has a particular energy. Each orbital can hold a maximum of two electrons. The number of orbitals in each sublevel is fixed:

Number of orbitals and electron capacity for each sublevel.

Degenerate orbitals are orbitals in the same sublevel with the same energy. The three p orbitals in one p sublevel are degenerate; the five d orbitals in one d sublevel are degenerate too.

An orbital diagram uses arrows in boxes to show electron occupancy. Each box represents one orbital, and each arrow represents one electron. The relative vertical position of boxes can show relative energy. Up and down arrows show opposite spin; don’t picture literal spinning balls, but do use the notation correctly.

The Pauli exclusion principle says that an orbital can contain at most two electrons and, if two are present, they must have opposite spins. In an orbital diagram, a full orbital is written as ↑↓, not ↑↑.

The Hund’s rule says that electrons fill degenerate orbitals singly, with parallel spins, before any pairing starts. For example, a p³ arrangement is ↑ ↑ ↑ across the three p boxes, not ↑↓ ↑ with one empty p orbital. This lowers electron–electron repulsion and gives a more stable arrangement.

The Aufbau principle says that electrons go into the lowest available energy orbitals before higher-energy orbitals are occupied. For atoms up to atomic number 36, use this filling order:

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p

Here atomic number, Z, is the number of protons in the nucleus of an atom (unitless). In a neutral atom, the number of electrons is also Z.

Writing configurations up to Z = 36

A full electron configuration is an electron configuration written from 1s onwards, with every occupied sublevel shown. Calcium has Z = 20, so its full configuration is:

1s² 2s² 2p⁶ 3s² 3p⁶ 4s²

A condensed electron configuration is an electron configuration that replaces the inner electrons with the symbol of the previous noble gas in square brackets. For calcium, this is:

[Ar] 4s²

You are expected to know both forms. The condensed version is not a way to skip the thinking; it separates the noble-gas core from the outer electrons.

For ions, start by working out the number of electrons. A positive ion has lost electrons; a negative ion has gained electrons. For main-group ions, remove or add electrons until you reach the ion’s electron count, then write the configuration. For transition-metal ions in this syllabus range, keep one key rule in mind: when forming cations, 4s electrons are removed before 3d electrons. So Fe is [Ar] 4s² 3d⁶, but Fe²⁺ is [Ar] 3d⁶, not [Ar] 4s² 3d⁴.

Chromium and copper

Chromium and copper are the two ground-state exceptions you need to know. From Aufbau, you would expect Cr: [Ar] 4s² 3d⁴ and Cu: [Ar] 4s² 3d⁹. The observed configurations are:

• Cr: [Ar] 4s¹ 3d⁵
• Cu: [Ar] 4s¹ 3d¹⁰

This happens because half-filled and completely filled d sublevels have extra stability. In chromium, 3d⁵ gives five singly occupied d orbitals. In copper, 3d¹⁰ gives a full d sublevel. If copper forms Cu²⁺, remove the 4s electron first and then one 3d electron, giving [Ar] 3d⁹.

First ionization energy and the convergence limit

First ionization energy is the energy needed to remove one electron from each atom in one mole of gaseous atoms, forming one mole of gaseous 1+ ions. It is usually reported in kJ mol⁻¹. The process is:

X(g) → X⁺(g) + e⁻

The convergence limit is the high-frequency end of a set of spectral lines, where the lines sit so close together that they merge. In an emission spectrum, this limit matches ionization: the electron has moved from a bound energy level into the continuum, where the atom no longer holds it.

If you know the convergence wavelength or frequency, you can calculate the first ionization energy. For one photon, use E = hf. If wavelength is given, combine this with c = λf to give E = hc / λ.

To convert the energy for one atom into the energy for one mole, use IE₁ = ENₐ / 1000, where IE₁ is the first ionization energy (kJ mol⁻¹) and Nₐ is the Avogadro constant (mol⁻¹). Dividing by 1000 changes J mol⁻¹ into kJ mol⁻¹.

A typical calculation route is:

1. If given λ, calculate f using c = λf, or go directly to E = hc / λ.
2. Calculate the energy for one atom in J.
3. Multiply by Nₐ to get J mol⁻¹.
4. Divide by 1000 to get kJ mol⁻¹.

Trends across periods and down groups

First ionization energy generally increases across a period. The number of protons increases across the period, while added electrons go into the same main energy level. Shielding changes only slightly, so the nucleus attracts the outer electron more strongly and that electron becomes harder to remove.

First ionization energy generally decreases down a group. The outer electron is in a higher main energy level and sits farther from the nucleus. Inner electrons also provide more shielding. This extra distance and shielding outweigh the higher nuclear charge, so less energy is needed to remove the outer electron.

These trends help explain the broad metal–non-metal pattern in the periodic table. Elements with low first ionization energies lose electrons more readily and tend to show metallic behaviour. Elements with high first ionization energies hold onto electrons more strongly and tend to show non-metallic behaviour.

The important discontinuities

There are two standard dips across a period that you need to explain from electron configurations.

Between group 2 and group 13, the first ionization energy drops. For example, Be is 1s² 2s², while B is 1s² 2s² 2p¹. Boron loses a 2p electron, which is slightly higher in energy and more shielded than a 2s electron, so it is easier to remove.

Between group 15 and group 16, the first ionization energy also drops. For example, N is 1s² 2s² 2p³, while O is 1s² 2s² 2p⁴. Nitrogen has three singly occupied p orbitals; oxygen has one paired set in a p orbital. Repulsion between the paired electrons in oxygen makes one electron easier to remove.

Log scales are useful when ionization energies span a large range, just as they are useful for hydrogen ion concentration. A logarithmic plot compresses very large values while keeping the patterns, so jumps and trends can be compared more clearly on one graph.