Electron configurations

📚The chemistry

Light, and what emission spectra tell us

Across the electromagnetic spectrum, wavelength and frequency are inversely related, and energy is proportional to frequency:

Both equations, and the values of \( c \) and \( h \), are in the data booklet. Together they say that short wavelength means high frequency means high energy. So across the visible region, red light (long \( \lambda \)) is the lowest energy and violet the highest; beyond violet lies ultraviolet, and beyond red lies infrared.

A continuous spectrum contains all wavelengths merging into one another — what a hot filament or the sun produces. A line spectrum contains only certain discrete wavelengths, separated by darkness — what an excited gaseous element produces.

An emission spectrum arises when electrons are promoted to higher energy levels by heat or electricity, and then fall back. Each fall releases a photon whose energy is exactly the difference between the two levels, \( \Delta E = hf \). Because only certain energy levels exist, only certain photon energies are possible, so only certain lines appear.

The hydrogen line spectrum

Hydrogen has one electron, so its spectrum is the simplest and is the one you must be able to describe. The lines fall into series, each series being the set of transitions ending at a particular level. The names of the series are explicitly not assessed, but the levels are:

• transitions down to \( n = 1 \) release the most energy and lie in the ultraviolet;
• transitions down to \( n = 2 \) lie in the visible region — these are the four coloured lines you see through a spectroscope;
• transitions down to \( n = 3 \) lie in the infrared.

Within any series the lines converge at higher frequency. That is the crucial observation: the energy levels themselves get closer together as \( n \) increases, so successive transitions differ by less and less. At the convergence limit the lines merge and the spectrum becomes continuous, which corresponds to \( n = \infty \) — the electron has left the atom entirely. That is ionization, and it is the basis of the AHL calculation below.

Energy levels, sublevels and orbitals

The main energy level is labelled by an integer \( n \), and holds at most

Each main level is divided into sublevels of successively higher energy, labelled s, p, d and f. Level \( n \) contains \( n \) sublevels: \( n=1 \) has 1s only; \( n=2 \) has 2s and 2p; \( n=3 \) has 3s, 3p and 3d; \( n=4 \) adds 4f.

Each sublevel contains a fixed number of orbitals — regions of space in which there is a high probability of finding an electron — and each orbital holds at most two electrons, of opposite spin.

You need to recognise the shape and orientation of an s orbital and of the three p orbitals; the shapes of d and f orbitals are outside the syllabus.

Writing electron configurations

Three rules, and you should be able to name all three:

• Aufbau principle — electrons fill the lowest-energy orbital available first.
• Pauli exclusion principle — an orbital holds at most two electrons, and they must have opposite spin.
• Hund’s rule — within a sublevel, electrons occupy orbitals singly, with parallel spins, before any orbital is doubled up. (Two electrons in the same orbital repel each other, so spreading out is lower in energy.)

The filling order is not simply by \( n \), because the 4s sublevel is lower in energy than 3d when both are empty:

So iron, \( Z = 26 \), is \( 1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2\,3d^6 \), or condensed using the preceding noble gas core, \( [\mathrm{Ar}]\,4s^2\,3d^6 \). Both forms are accepted; the condensed form is faster and less error-prone.

Two exceptions must be learned, chromium and copper, because a half-filled or completely filled d sublevel is a particularly stable arrangement:

Ions of transition elements lose 4s electrons first. This surprises people, since 4s filled first, but once 3d is occupied the 4s electrons are the higher in energy and the outermost. So \( \mathrm{Fe}^{2+} \) is \( [\mathrm{Ar}]\,3d^6 \) and \( \mathrm{Fe}^{3+} \) is \( [\mathrm{Ar}]\,3d^5 \) — a half-filled d sublevel, which is part of why iron(III) is so stable. For main-group ions, remove from or add to the outermost p sublevel as usual.

AHL Ionization energy from the convergence limit

The first ionization energy is the energy needed to remove one mole of electrons from one mole of gaseous atoms:

The convergence limit of the \( n = 1 \) series in the hydrogen spectrum corresponds to the transition from \( n = \infty \) to \( n = 1 \) — that is, to ionization. So if you are given the frequency or wavelength at the convergence limit, you can calculate the ionization energy:

1. if given \( \lambda \), find \( f \) from \( c = \lambda f \);
2. find the energy of one photon from \( E = hf \), in joules;
3. multiply by the Avogadro constant to get energy per mole;
4. divide by 1000 to convert J mol⁻¹ to kJ mol⁻¹.

Steps 3 and 4 are where the marks are lost. An ionization energy is a molar quantity; a photon energy is not.

AHL Trends and discontinuities in first ionization energy

Down a group, first ionization energy falls. The outer electron is in a higher main energy level, so it is further from the nucleus and there are more inner shells shielding it. The attraction is weaker and the electron is easier to remove. (Nuclear charge does increase down a group, but the increased distance and shielding outweigh it.)

Across a period, first ionization energy generally rises. The nuclear charge increases while electrons are added to the same main level, so shielding stays essentially constant. Effective nuclear charge rises, atomic radius falls, and the outer electron is held more tightly.

But the rise is not smooth, and the two discontinuities in period 3 are direct evidence that sublevels exist:

• Al is lower than Mg. The electron removed from aluminium comes from a 3p orbital, which is higher in energy than the 3s orbital from which magnesium’s is removed. A 3p electron is on average further out and slightly shielded by the 3s pair, so less energy is needed.
• S is lower than P. In phosphorus, \( 3p^3 \), each p orbital holds one electron. In sulfur, \( 3p^4 \), one orbital holds a pair, and the two electrons in that orbital repel each other, making one easier to remove.

AHL Successive ionization energies

Remove electrons one after another and every value is larger than the last, because you are pulling a negative charge away from an increasingly positive ion. What matters is where the large jump occurs: it marks the point at which the next electron must come from an inner main energy level, much closer to the nucleus and much less shielded.

The number of electrons removed before the big jump is the number of outer-shell electrons, which gives the group. Remove two easily and then hit a wall, and the element is in group 2. A plot of \( \log(\mathrm{IE}) \) against electron number makes the jumps visible on one graph — logarithms are used because the values span several orders of magnitude.