The electronic configuration of gadolinium (Atomic number 64) is (i) [Xe] 4f³ 5d⁵ 6s² (ii) [Xe] 4f⁷ 5d² 6s¹ (i

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Q. The electronic configuration of gadolinium (Atomic number 64) is

(i) [Xe] 4f^3 5d^5 6s^2

(ii) [Xe] 4f^7 5d^2 6s^1

(iii) [Xe] 4f^7 5d^1 6s^2

(iv) [Xe] 4f^8 5d^6 6s^2

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Concept understanding — Aufbau Principle

The Aufbau Principle: Building Atoms from the Ground Up

Imagine you're building a house. You don't start with the roof — you lay the foundation first, then the walls, then the roof. Electrons in an atom behave the same way: they fill the lowest-energy "rooms" (orbitals) before moving to higher-energy ones. That's the core idea behind the Aufbau Principle.

The word Aufbau comes from German, meaning "building up" or "construction." It describes how electrons are added one by one to an atom as you move across the periodic table.

The Intuition: Why Lowest Energy First

An atom is most stable when its electrons are in the lowest possible energy states. Think of it like a ball rolling down a hill — it naturally settles at the bottom, not halfway up. If you try to put an electron into a high-energy orbital while a lower-energy orbital is empty, the atom would be unstable and the electron would "fall" down to the empty spot, releasing energy.

So nature follows the path of least energy: fill from the bottom up.

The Precise Statement

Aufbau Principle: In the ground state of an atom, electrons fill atomic orbitals in order of increasing orbital energy. Lower-energy orbitals are completely filled before electrons occupy higher-energy orbitals.

This is not a guess — it's a rule that emerges from quantum mechanics and is confirmed by experimental data (like atomic spectra and ionization energies).

The Order of Filling: The (n + l) Rule

But here's where it gets interesting: orbital energies don't simply increase with principal quantum number $n$ . For example, the 4s orbital ( $n=4, l=0$ ) actually has lower energy than the 3d orbital ( $n=3, l=2$ ). So electrons fill 4s before 3d.

To predict the correct order, use the (n + l) rule:

Tip

For an orbital with quantum numbers $n$ and $l$ , its energy increases with $(n + l)$ . If two orbitals have the same $(n + l)$ value, the one with lower $n$ fills first.

Let's check:

4s: $n + l = 4 + 0 = 4$
3d: $n + l = 3 + 2 = 5$

Since $4 < 5$ , 4s fills first. That's why potassium (atomic number 19) has the electron configuration $[\text{Ar}]\,4s^1$ , not $[\text{Ar}]\,3d^1$ .

The Complete Filling Order (Memory Aid)

Here's the sequence you'll use for most elements up to atomic number 118:

1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow 4p \rightarrow 5s \rightarrow 4d \rightarrow 5p \rightarrow 6s \rightarrow 4f \rightarrow 5d \rightarrow 6p \rightarrow 7s \rightarrow 5f \rightarrow 6d \rightarrow 7p

A common trick is to write the orbitals in rows by $n$ and draw diagonal arrows:

1s
2s  2p
3s  3p  3d
4s  4p  4d  4f
5s  5p  5d  5f
6s  6p  6d
7s  7p

Follow the diagonals from top-right to bottom-left — that gives you the filling order.

Important Exceptions (Don't Panic)

The Aufbau principle works beautifully for most elements, but there are a few well-known exceptions where a half-filled or fully-filled subshell provides extra stability:

Chromium ( $Z=24$ ): Expected $[\text{Ar}]\,4s^2 3d^4$ , but actual is $[\text{Ar}]\,4s^1 3d^5$ (half-filled d-subshell is more stable) …

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Gadolinium achieves extra stability through a half-filled $4f^7$ subshell combined with the minimum occupancy needed in higher orbitals. The configuration is [Xe] $4f^7$ $5d^1$ $6s^2$ .

The electronic configuration of lanthanides is governed by the interplay between orbital energies and the exceptional stability of half-filled and fully-filled subshells. For gadolinium, this stability principle creates a striking departure from the naive filling order.

Why half-filled $f$ subshells matter

The $4f$ subshell can hold 14 electrons across seven orbitals. When exactly half-filled ( $4f^7$ ), each of the seven orbitals contains one electron with parallel spin. This arrangement minimizes electron-electron repulsion (exchange energy is maximized) and creates unusual stability—similar to the half-filled $d^5$ configuration you see in chromium and manganese.

For lanthanides around the middle of the series, the energy cost of promoting an electron from $6s$ or $5d$ into $4f$ can be offset by the stability gained from achieving $4f^7$ .

Working through gadolinium's configuration

Start with xenon core: Gadolinium (atomic number 64) has 64 electrons. The xenon core [Xe] accounts for 54 electrons, leaving 10 more to place.
Expected filling order: Naively, after [Xe] we fill $4f$ before $5d$ (since $4f$ is lower in energy for lanthanides). We'd expect to place electrons as $6s^2$ first (2 electrons), then $4f^8$ (8 electrons), giving [Xe] $4f^8$ $6s^2$ .
Stability check: But $4f^8$ is one electron past the half-filled $4f^7$ configuration. The system can gain significant exchange energy by keeping $4f^7$ instead.
Redistribution: To achieve $4f^7$ while accounting for all 10 electrons beyond xenon:
- Place $4f^7$ (7 electrons) — half-filled, maximum stability
- Place $6s^2$ (2 electrons) — the $6s$ orbital fills early and remains occupied
- Place $5d^1$ (1 electron) — the "extra" electron goes here rather than making $4f^8$ …

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Q.The electronic configuration of gadolinium (Atomic number 64) is

Why half-filled fff subshells matter

Working through gadolinium's configuration

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Why half-filled $f$ subshells matter

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