Electron configurations

When atoms are given energy (by heating or an electric discharge), electrons are promoted to higher energy levels. They are unstable there and quickly fall back to lower levels, releasing the energy difference as a photon of light.

Because energy levels are quantized (only certain values are allowed), only photons of certain fixed energies can be emitted. This produces a line emission spectrum - sharp coloured lines on a black background - rather than a continuous rainbow.

The photon energy is fixed by the gap between the two levels:

$$E = hf = \frac{hc}{\lambda}$$

so a larger energy gap gives a higher frequency (shorter wavelength, towards the blue/violet end).

Key features of the hydrogen spectrum:

• Lines appear in series (e.g. the visible Balmer series, where electrons fall to $n=2$).
• Within each series the lines get closer together (converge) at higher frequency.
• The convergence happens because the energy levels themselves get closer together as $n$ increases.

The existence of discrete lines is the central piece of evidence that electrons occupy discrete energy levels, not a continuous range of energies. The booklet provides $c = f\lambda$ and $E = hf$ for these calculations.

Each main energy level (shell), numbered by the principal quantum number $n = 1, 2, 3, ...$, is divided into sublevels (subshells) labelled s, p, d, f.

The number of sublevels equals $n$:

• $n = 1$: 1s only
• $n = 2$: 2s, 2p
• $n = 3$: 3s, 3p, 3d
• $n = 4$: 4s, 4p, 4d, 4f

Each sublevel holds a fixed maximum number of electrons:

• s holds 2 (1 orbital)
• p holds 6 (3 orbitals)
• d holds 10 (5 orbitals)
• f holds 14 (7 orbitals)

The sublevels are not filled strictly by $n$. Their energies overlap, and the order of filling is:

$$1s\ 2s\ 2p\ 3s\ 3p\ 4s\ 3d\ 4p\ 5s\ 4d\ 5p\ ...$$

The key crossover is that 4s is lower in energy than 3d, so 4s fills first. This explains why the d-block (transition metals) starts in period 4. Sublevels within the same level rise in energy in the order s < p < d < f.

An electron configuration shows how electrons are distributed among sublevels, e.g. iron is $1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2\,3d^6$. Three rules govern the filling:

• Aufbau principle: fill the lowest-energy sublevel available first (follow the order $1s, 2s, 2p, 3s, 3p, 4s, 3d...$).
• Pauli exclusion principle: an orbital holds a maximum of 2 electrons, and they must have opposite spins (drawn as $\uparrow\downarrow$).
• Hund's rule: within a set of equal-energy (degenerate) orbitals, electrons occupy them singly with parallel spins first, then pair up. This minimizes repulsion.

Condensed notation uses the previous noble gas, e.g. $[\text{Ar}]\,4s^2\,3d^6$ for iron.

Forming ions: when transition metals form cations, electrons are removed from 4s before 3d (e.g. $\text{Fe}^{2+}$ is $[\text{Ar}]\,3d^6$).

Two anomalies to learn:

• Chromium: $[\text{Ar}]\,4s^1\,3d^5$ (not $4s^2\,3d^4$)
• Copper: $[\text{Ar}]\,4s^1\,3d^{10}$ (not $4s^2\,3d^9$)

A half-filled ($3d^5$) or fully filled ($3d^{10}$) d sublevel gives extra stability.

An atomic orbital is a region where an electron is most likely to be found. Each orbital holds at most 2 electrons with opposite spins.

s orbitals are spherical and centred on the nucleus. There is one s orbital in each level (1s, 2s, 3s...). Higher s orbitals are larger.

p orbitals are dumbbell-shaped (two lobes either side of the nucleus). There are three p orbitals in any p sublevel, identical in shape and energy but pointing along the three perpendicular axes:

• $p_x$ along the x-axis
• $p_y$ along the y-axis
• $p_z$ along the z-axis

Because the three p orbitals are degenerate (equal energy), Hund's rule applies when filling them: they each take one electron before any pairs up.

Useful summary of capacities:

• s sublevel: 1 orbital, 2 e$^-$
• p sublevel: 3 orbitals, 6 e$^-$
• d sublevel: 5 orbitals, 10 e$^-$

For IB you only need to know the shapes of s and p orbitals; you do not need the shapes of d or f orbitals. An orbital diagram uses a box (or line) per orbital and arrows for electron spins.

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

$$\text{X(g)} \rightarrow \text{X}^+\text{(g)} + e^-$$

It is always endothermic (positive). Ionization energy gives direct evidence for electron arrangement.

Link to emission spectra: in the hydrogen spectrum, the lines of a series get closer together at higher frequency and eventually converge to a limit. This convergence limit corresponds to the electron jumping from $n=1$ all the way to $n=\infty$ - that is, leaving the atom completely. So the energy of the photon at the convergence limit equals the ionization energy of one atom.

To find the ionization energy per mole:

• Read off the frequency at the convergence limit, $f_{\text{limit}}$.
• Energy of one photon: $E = hf_{\text{limit}}$.
• Multiply by the Avogadro constant: $E_{\text{mol}} = N_A\,hf_{\text{limit}}$.
• Convert from $\text{J mol}^{-1}$ to $\text{kJ mol}^{-1}$.

Trends to know: ionization energy generally increases across a period (rising nuclear charge) and decreases down a group (electrons further from the nucleus, more shielding). Small dips (e.g. from group 2 to 13, and group 15 to 16) give evidence for sublevels and orbital pairing.

Successive ionization energies are the energies to remove the 1st, 2nd, 3rd... electron from the same atom, one after another:

$$\text{X(g)} \rightarrow \text{X}^+ \rightarrow \text{X}^{2+} \rightarrow \text{X}^{3+} ...$$

They always increase for a given element, because:

• Each electron is pulled from an increasingly positive ion, so it is held more strongly.
• The remaining electrons feel less repulsion and are drawn closer.

The most useful feature is the large jumps. A sudden big rise occurs when an electron must be removed from a new, inner main energy level that is closer to the nucleus and less shielded.

Reading the data:

• Plot $\log(\text{ionization energy})$ against electron number to make the jumps clear.
• Count the electrons removed before the first big jump - this equals the number of electrons in the outer shell, i.e. the group number.
• The pattern of jumps confirms the shell structure (2, 8, 8...).

For example, sodium shows a huge jump after the 1st electron, confirming 1 outer electron (group 1). Magnesium jumps after the 2nd (group 2). This data is strong evidence for the existence of discrete energy levels.