S1.3 Electron configurations

Evidence for energy levels

When atoms are given energy (heat, electric discharge) electrons jump to higher levels. When they fall back, they release the energy difference as a photon:

\[\Delta E = hf = \frac{hc}{\lambda}\]

Because only certain energy gaps exist, only certain frequencies are emitted, giving a line spectrum of separate coloured lines on a dark background. A continuous spectrum (like white light through a prism) contains all wavelengths. Line spectra are the evidence that electron energy is quantised.

The hydrogen spectrum

• Transitions falling to n = 1 release the most energy: ultraviolet lines.
• Transitions falling to n = 2 give the visible lines.
• Transitions to n = 3 give infrared lines.

Within each series the lines get closer together (converge) at higher energy because energy levels get closer together further from the nucleus. The convergence limit corresponds to the electron being removed completely.

Sublevels and orbitals

• Main energy level n holds up to \(2n^2\) electrons.
• Sublevels: s (1 orbital), p (3 orbitals), d (5 orbitals), f (7 orbitals).
• Each orbital holds a maximum of 2 electrons with opposite spins.
• s orbitals are spherical; p orbitals are dumbbell-shaped along x, y and z.

Filling rules

1. Aufbau principle: fill the lowest-energy sublevel first: 1s 2s 2p 3s 3p 4s 3d 4p.
2. Pauli exclusion principle: no more than two electrons per orbital, with opposite spins.
3. Hund's rule: in a set of equal-energy orbitals, electrons occupy separate orbitals singly with parallel spins before pairing.

Examples:

• Si (Z = 14): 1s\(^2\) 2s\(^2\) 2p\(^6\) 3s\(^2\) 3p\(^2\), or [Ne] 3s\(^2\) 3p\(^2\)
• Fe (Z = 26): [Ar] 3d\(^6\) 4s\(^2\)
• Exceptions: Cr is [Ar] 3d\(^5\) 4s\(^1\) and Cu is [Ar] 3d\(^{10}\) 4s\(^1\) (half-full and full d sublevels are especially stable).
• Transition metal ions lose 4s electrons first: Fe\(^{3+}\) is [Ar] 3d\(^5\).

Ionisation energy from spectra (HL)

The convergence limit of the series ending at n = 1 gives the energy needed to remove the electron from the ground state.

Worked example (HL)

The hydrogen series ending at n = 1 converges at 91.2 nm. Find the first ionisation energy of hydrogen in kJ mol\(^{-1}\).

\[E = \frac{hc}{\lambda} = \frac{(6.63 \times 10^{-34}\ \text{J s})(3.00 \times 10^{8}\ \text{m s}^{-1})}{91.2 \times 10^{-9}\ \text{m}} = 2.18 \times 10^{-18}\ \text{J per atom}\]

\[2.18 \times 10^{-18}\ \text{J} \times 6.02 \times 10^{23}\ \text{mol}^{-1} = 1.31 \times 10^{6}\ \text{J mol}^{-1} = 1.31 \times 10^{3}\ \text{kJ mol}^{-1}\]

Successive ionisation energies (HL)

Each successive ionisation energy is larger, because each electron is removed from an increasingly positive ion. A large jump shows the next electron comes from a shell closer to the nucleus.

Example: an element has successive values 738, 1451, 7733 and 10 540 kJ mol\(^{-1}\). The big jump is between the 2nd and 3rd, so there are 2 outer electrons: the element is in group 2 (this is magnesium).

Common mistakes

• Writing Fe\(^{2+}\) as [Ar] 3d\(^4\) 4s\(^2\) instead of [Ar] 3d\(^6\).
• Forgetting the Cr and Cu exceptions.
• Saying the lines converge at low energy: they converge at high energy (short wavelength).
• Pairing electrons in p or d orbitals before every orbital has one electron.
• Leaving wavelength in nm in \(E = hc/\lambda\).

Exam tips

• For "how many unpaired electrons", draw orbital boxes; it takes ten seconds and avoids errors.
• Describe emission as electrons falling from higher to lower levels and emitting photons of specific energy.
• In ionisation energy data, look for the jump, count the electrons before it, and that is the group number (for main-group elements).
• Convert J per atom to kJ per mole by multiplying by \(N_A\) and dividing by 1000.