Electron Configuration: Aufbau Principle, 4s vs 3d, and Orbital Filling Order

Context from Transcript

• The transcript fragment mentions the orbital filling sequence: 2p, 3s, 3p, to 4s, and then asks what comes next, suggesting 3d.

• The core question inferred: Why does the filling order go from 4s before 3d?

• This is a common point of confusion in electron configurations and relates to how orbitals are ordered by energy in multi-electron atoms.

Key Concept: Orbital Filling Order (Aufbau Principle)

• Electrons populate atomic orbitals to minimize energy, following a nearly universal sequence.

• Observed approximate filling order (from the transcript context):

  • 2p, 3s, 3p, 4s, 3d, …

• Goal: Understand why 4s is filled before 3d and what governs the subsequent steps.

Main Rule: Madelung (n + l) Rule

• Electrons fill orbitals in order of increasing value of $n + l$, where:

  • $n$ = principal quantum number

  • $l$ = azimuthal quantum number (s: 0, p: 1, d: 2, f: 3)

• If two orbitals have the same $n + l$ value, the orbital with the smaller $n$ fills first.

• Practical consequence: 4s (n=4, l=0, so $n+l=4$) fills before 3d (n=3, l=2, so $n+l=5$).

• This provides the primary explanation for the initial sequence: E(1s) < E(2s) < E(2p) < E(3s) < E(3p) < E(4s) < E(3d) < E(4p) < \,\ldots

• Note: In real atoms, electron–electron interactions slightly perturb energies, so the ordering is approximate and can shift for certain elements.

Supporting Physics: Quantum Numbers and Orbital Types

• Orbital types by angular momentum quantum number $l$:

  • $s <br>ightarrow l=0$

  • $p <br>ightarrow l=1$

  • $d <br>ightarrow l=2$

  • $f <br>ightarrow l=3$

• Each orbital can hold up to two electrons with opposite spins (Pauli exclusion principle):

  • Maximum occupancy per orbital: $2$ with spins $m<em>s = +\frac{1}{2}$ and $m</em>s = -\frac{1}{2}$.

• Hund's rule (tie-breaking within a subshell): electrons fill degenerate orbitals singly with parallel spins before pairing.

Practical Rules and Notable Exceptions

• Aufbau principle (comprehensive version) uses the Madelung rule as the core guideline.

• Exceptions for stability in transition metals:

  • Chromium: $[Ar] 3d^5\, 4s^1$ (instead of $[Ar] 3d^4\, 4s^2$)

  • Copper: $[Ar] 3d^{10}\, 4s^1$ (instead of $[Ar] 3d^9\, 4s^2$)

• Why these exceptions? Half-filled and fully-filled d subshells offer extra stability due to exchange energy and electron–repulsion considerations, lowering the overall energy.

Step-by-Step Fill for Early Elements (Illustrative Example)

• Process example up to calcium (for context):

  • 1s^2, 2s^2 2p^6, 3s^2 3p^6, 4s^2, 3d^0 (before filling) — then 4p, etc., as we move through the periodic table.

• This sequence aligns with the order: $1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow 4p \rightarrow 5s \rightarrow \text{etc.}$

Mathematical Details and Notation

• Key quantum numbers and notations:

  • Principal quantum number: $n = 1, 2, 3, \dots$

  • Azimuthal quantum number: $l = 0, 1, 2, 3, \dots\quad (s, p, d, f)$

  • Magnetic quantum number: $m_l = -l, -l+1, \dots, l$

  • Spin quantum number: $m_s = \pm \frac{1}{2}$

• Energy ordering approximation (Madelung rule):

  • $E(n,l) \text{ increases with } n+l,$ and for equal $n+l$, the smaller $n$ has lower energy.

  • Therefore, the filling sequence begins with orbitals that minimize $n+l$ and respects the tie-breaker on $n$.

• Orbital capacity:

  • Each orbital can hold up to 2 electrons: total capacity of a subshell is $2(2l+1)$ where the number of orbitals in the subshell is $2l+1$.

Why the Question Matters: Significance for Chemistry and Real-World Relevance

• Electron configuration determines valence electron arrangement, which drives chemical properties, bonding, and periodic trends.

• The order (2p, 3s, 3p, 4s, 3d, …) explains why elements in certain groups have similar valence structures and why the chemistry changes across periods.

• Understanding exceptions helps explain anomalous behaviors (e.g., Cr and Cu in the first row of the 3d series).

Quick Summary (Exam-Oriented Takeaways)

• The initial filling order in many textbooks is: $1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow 4p \rightarrow \cdots$

• The primary reason 4s fills before 3d is that

  • n+l(4s) = 4+0 = 4 < n+l(3d) = 3+2 = 5,

  • hence 4s is energetically lower than 3d in the Aufbau (Madelung) ordering.

• Exceptions like chromium and copper illustrate that energy is not a simple monotonic function of n; electronic stability can favor unusual configurations:

  • $\text{Cr}: [Ar] 3d^5\, 4s^1$

  • $\text{Cu}: [Ar] 3d^{10}\, 4s^1$

• Core principles to memorize: Pauli exclusion, Hund's rule, and the Aufbau/Madelung rule with the n+l framework.