Chapter 1 built the seats. Here you fill them. You will play the seating game by hand, then watch the seat heights change as $Z$ grows, and see why 4s fills before 3d in potassium yet leaves first when iron rusts.
2.1 Two electrons per orbital
Every electron carries a built-in property called spin. Spin is not a literal rotation; it is a quantum number that takes exactly two values, written $\uparrow$ and $\downarrow$. The Pauli exclusion principle says no two electrons in one atom can share the exact same quantum state. An orbital already fixes the quantum numbers $(n, \ell, m)$, so spin is the only freedom left. Two electrons can share an orbital only if one is $\uparrow$ and the other is $\downarrow$.
That gives a hard cap of two electrons per orbital. Counting orbitals per subshell gives the seat totals: one s orbital holds 2 electrons, three p orbitals hold 6, five d hold 10, seven f hold 14. Those numbers are the widths of the blocks in the periodic table.
2.2 Shielding breaks the tie
In a hydrogen atom (one proton, one electron) the energy of an orbital depends only on the shell number $n$. The 2s and 2p subshells have the exact same energy. When two states share an energy like that, physicists call them degenerate.
Add more electrons and that degeneracy breaks. An outer electron in a multi-electron atom does not feel the full pull of the nucleus, because the inner electrons sit between it and the nucleus and cancel part of that pull. The cancellation is called shielding, and what the outer electron actually feels is a reduced, effective nuclear charge:
$$Z_\mathrm{eff} = Z - \htmlData{tip=The shielding constant - how much pull the inner electrons cancel}{S}$$
Here $Z$ is the proton count and $S$ is how much of that pull the inner electrons block. Now chapter 1's penetration idea pays off: orbitals with more probability close to the nucleus slip past the inner cloud and feel more of the bare charge. An s orbital dips deepest and feels nearly the full nuclear pull; a p orbital dips less; a d orbital barely penetrates at all and sits mostly outside the screen. So $Z_\mathrm{eff}$ is bigger for s than p than d, and a bigger $Z_\mathrm{eff}$ means a more tightly bound, lower-energy electron.
$E_{ns} < E_{np} < E_{nd} < E_{nf}$. The more an orbital penetrates, the more nucleus it feels, and the lower its energy.
Why s sees more nucleus than d
An outer s electron pokes inside the inner-electron cloud and feels the bare nucleus. A d electron stays outside and feels a screened, weaker pull.
2.3 The seating rules
Three rules, taken together, decide where every electron in a ground-state atom ends up.
- Aufbau (German for "building up"): each new electron takes the lowest-energy seat that is still open.
- Pauli: at most two electrons per orbital, and those two must have opposite spins.
- Hund: when a subshell has several orbitals at the same energy (the three p orbitals, the five d orbitals), electrons spread out into separate orbitals before doubling up. Pairing two electrons in one orbital costs energy because they repel each other, so unpaired same-spin arrangements come first.
Build an atom, one electron at a time
Pick a target element, then click an orbital to drop in the next electron. Misplace one and the rules will say which one you broke.
2.4 The rungs move
The energy ladder you just used looks fixed, but it is not. Each subshell has its own energy, and that energy depends on $Z$: the more protons the nucleus has, the harder it pulls every electron in. Different subshells respond to that pull at different rates. The 4s curve dips gently as $Z$ grows; the 3d curve dips much faster, especially once 3d starts to fill. Somewhere between potassium ($Z=19$) and scandium ($Z=21$), the two curves cross.
The energies below come from Slater's rules, an empirical recipe for $Z_\mathrm{eff}$ that John Slater published in 1930. Real atoms need numerical Hartree–Fock to get the energies exactly right, but Slater's rules catch the crossings and trends honestly on one page of arithmetic. Treat the chart as a faithful model of a real effect, not the last word.
Subshell energy vs atomic number
Drag the slider to change $Z$. Watch the 4s and 3d lines: they cross between Ca and Sc, and the filling order flips at the crossing.
The aufbau order is not a list to memorize. Each electron takes the cheapest open seat at the moment it arrives, and the seats themselves move as $Z$ grows. At $Z=19$ the cheapest open seat is 4s, so potassium's last electron lands there. By $Z=21$ the cheapest open seat is 3d. Same rule, different answer.
2.5 Last in, first out — except
Here is a fact that confuses chemistry students every year. When iron forms the Fe2+ ion, the two electrons it loses come from 4s, not from 3d. Ionization means pulling electrons off an atom, and an atom gives up its highest-energy electrons first because those are the easiest to take. But we just said 4s fills before 3d — filled first, yet emptied first. The chart resolves the contradiction.
Look back at the chart from 2.4 near iron ($Z=26$). The 3d curve has plunged so far by then that 3d sits well below 4s; the 3d electrons are now core-like, locked in tight. The 4s pair are the highest-energy electrons in neutral iron, so they are the ones ionization peels off first. 4s was the cheapest seat available when iron was filling, and the most expensive electron to hold onto once iron was full.
Ionize a transition metal
Pick a neutral atom, then remove two electrons and watch which seat empties first.
2.6 The exceptions prove the rule
Two elements in the first transition row refuse to follow the seating game:
- Chromium ($Z=24$): not [Ar] 3d4 4s2, but [Ar] 3d5 4s1.
- Copper ($Z=29$): not [Ar] 3d9 4s2, but [Ar] 3d10 4s1.
Near $Z=24$ and $Z=29$, the 4s and 3d energies sit nearly equal, so a small effect can break the tie. That small effect is electron repulsion: electrons in separate orbitals stay farther apart than electrons paired in the same orbital, which lowers the atom's total energy. A half-filled d subshell (five electrons, one in each d orbital, all same spin) and a completely filled d subshell (ten electrons) are both especially good at this. Promoting one electron from 4s up to 3d costs a little energy but gives chromium a half-filled d and copper a full d, and the repulsion saved beats the climb paid for.
Atoms minimize their total energy, not their energy one orbital at a time. The aufbau game from 2.3 is the right answer when the gap between subshells is large. When the gap shrinks, electron repulsion gets a vote, and once in a while that vote flips the result. The exceptions confirm that the underlying law is total-energy minimization. The seating chart is just its usual answer.
2.7 What you now have
- Pauli caps every orbital at two electrons of opposite spin.
- Shielding plus penetration breaks the hydrogen-only degeneracy: within a shell, $E_{ns} < E_{np} < E_{nd}$.
- Aufbau, Hund, and Pauli together predict the ground-state configuration of every atom you tried.
- Subshell energies move with $Z$. The 4s/3d crossing between Ca and Sc is why 4s fills first and why transition metals ionize from 4s.
- Cr and Cu deviate from the chart because the real law is total-energy minimization, not the chart itself.
Build the atom
Five elements, four candidate configurations each. Pick the ground state. Cr and Cu come last; the chart will not save you.