Aufbau principle — order of filling (Madelung rule, n + l): Question bank

Quick symbol refresher so nothing here is used unearned:

$n$ = principal quantum number = which "shell" (row) the orbital lives in ( $1, 2, 3, \dots$ ). Bigger $n$ → farther from the nucleus on average.
$l$ = azimuthal quantum number = the shape code of the orbital: $l = 0, 1, 2, 3$ means $s, p, d, f$ . See Quantum Numbers (n, l, m, s).
$K = n + l$ = the Madelung sum, the "price tag" that ranks filling order.

True or false — justify

TF1. "A 4s electron is always lower in energy than a 3d electron."

False. It's true for the empty (filling) atom, but once 3d starts filling, the extra nuclear pull drops 3d below 4s — which is exactly why 4s leaves first on ionization. Energy ordering is not fixed; it shifts with occupancy.

TF2. "Madelung's

n + l

rule is a fundamental law of nature."

False. It is a bookkeeping shortcut that packages the real physics of Penetration and Shielding. It works for predicting ground states but is not derived from a single equation, and it fails for a handful of atoms (Cr, Cu, …).

TF3. "Two orbitals can have the same

n + l

but different energies."

True. E.g. 3d, 4p, 5s all have

K = 5

yet differ in energy; the tie-break (smaller

n

first) exists precisely because equal

K

does not mean equal energy — inner shells penetrate more.

TF4. "The Aufbau order and the electron-removal order are the same list read backwards."

False. Filling ranks by

n + l

; removal ranks by highest

n

first. That's why 4s fills before 3d but empties before it too. See Ionization and Electron Removal Order.

TF5. "Increasing

l

at fixed

n

raises the energy."

True. Higher

l

means a more angular, less-penetrating shape, so it is shielded more and feels less nuclear charge → higher energy. This is half of what

n + l

encodes.

TF6. "Every element's ground state can be obtained by strict one-electron-at-a-time Aufbau."

False. Most can, but half-filled/full-filled subshell stability overrides Aufbau for cases like Cr and Cu. Aufbau gives the starting prediction, not a guarantee.

Spot the error

SE1. "Order: 1s 2s 2p 3s 3p 3d 4s …"

The error is placing 3d before 4s.

K_{4 s} = 4 < K_{3 d} = 5

, so 4s fills first: …3p 4s 3d 4p…

SE2. "

F e^{2 +} = [A r] 3 d^{4} 4 s^{2}

Wrong orbital emptied. Removal takes highest-

n

electrons first, so both 4s electrons leave:

F e^{2 +} = [A r] 3 d^{6}

SE3. "Cr

= [A r] 3 d^{4} 4 s^{2}

by plain Aufbau."

Aufbau's raw prediction is overridden. A half-filled

3 d^{5}

is extra stable, so Cr

= [A r] 3 d^{5} 4 s^{1}

SE4. "5s fills before 4d because

5 < ...

wait, larger

n

, so 4d wins."

Rank by

K

, not

n

K_{5 s} = 5 < K_{4 d} = 6

, so 5s fills first despite its larger

n

SE5. "s subshell holds 2, p holds 8, d holds 18."

Capacity is

2 (2 l + 1)

: s=2, p=6, d=10, f=14. The "8, 18" numbers are shell capacities (

2 n^{2}

), not subshell capacities. See Pauli Exclusion Principle.

SE6. "For 2p,

n + l = 2 + 2 = 4

For p,

l = 1

, not 2. So

K_{2 p} = 2 + 1 = 3

. (

l = 2

is d.)

SE7. "Since 3d and 4s are close in energy, order doesn't matter for the config of Fe."

Order matters for counting: filling 4s (2 e⁻) then 3d gives

3 d^{6}

, which sets Fe's chemistry and ion charge. Getting the crossover wrong changes the whole d-count.

Why questions

WHY1. Why isn't filling ordered by

n

alone?

Because in multi-electron atoms energy depends on both

n

and

l

(shape) through Penetration and Shielding; a penetrating 4s can dip below a diffuse 3d, breaking pure-

n

order.

WHY2. Why does the tie-breaker favor the smaller

n

Among orbitals with equal

n + l

, the one with smaller

n

is an inner shell that penetrates closer to the nucleus, feels more charge, and sits lower in energy — so it fills first.

WHY3. Why does 4s empty before 3d even though it filled first?

Once 3d is populated it drops below 4s in energy; ionization removes the now-highest-energy electron, which is the outer (higher-

n

) 4s.

WHY4. Why do half-filled and fully-filled subshells cause exceptions?

Symmetric electron distributions and favorable exchange energy lower the total energy enough to beat the Aufbau prediction, borrowing one electron from 4s into 3d (Cr, Cu).

WHY5. Why does the

n + l

rule literally reproduce the shape of the periodic table?

Each new

K

value adds the next block (s, then p, d, f) in sequence, so reading the Aufbau order is reading the table left-to-right, top-to-bottom. See Periodic Table Blocks (s, p, d, f).

WHY6. Why do we even need Hund's Rule on top of Aufbau?

Aufbau picks which subshell; Hund's rule decides how electrons arrange inside that subshell (spread out, spins parallel) to minimize repulsion. Different questions.

Edge cases

EC1. Hydrogen (

Z = 1

) has only one electron — does

n + l

ordering still apply?

For a one-electron atom energy depends on

n

alone (no shielding), so 2s and 2p are degenerate; the

n + l

subtlety only appears once multiple electrons shield each other.

EC2. What is

K = n + l

for the very first orbital, 1s?

K = 1 + 0 = 1

— the smallest possible

K

, which is why 1s always fills first.

EC3. Which fills first among orbitals that all have

K = 6

: 4d, 5p, 6s?

Tie-break by smaller

n

: 4d (

n = 3

... no,

n = 4

) < 5p (

n = 5

) < 6s (

n = 6

), so 4d < 5p < 6s.

EC4. Can two different orbitals ever have identical

n

and identical

l

No — that would make them the same subshell; they'd differ only by

m

or spin

s

, which are counted by capacity

2 (2 l + 1)

, not by Aufbau order.

EC5. Does the Madelung order ever put a higher-

K

orbital before a lower-

K

one?

Never for filling by the rule itself; the only "violations" are the true physical exceptions (Cr, Cu…) where subshell-stability energetics override the rule, not the rule reordering itself.

EC6. For a neutral atom vs its cation, is the "last orbital filled" the same as the "first orbital emptied"?

Generally no — 4s is the last filled but the first emptied, so the two questions have different answers whenever a d (or f) block is involved.

Connections

Quantum Numbers (n, l, m, s) — defines the $n$ and $l$ used in every trap here.
Penetration and Shielding — the physics that makes $n + l$ work.
Pauli Exclusion Principle — sets the $2 (2 l + 1)$ capacities behind SE5.
Hund's Rule — the "inside a subshell" question (WHY6).
Electron Configuration Exceptions (Cr, Cu) — the exceptions in TF6, SE3, WHY4.
Ionization and Electron Removal Order — the removal-order traps (TF4, SE2, WHY3).
Periodic Table Blocks (s, p, d, f) — WHY5's table connection.