Q.Using the $(n+l)$ rule, decide whether the $4s$ or the $3d$ subshell is filled first, and state which of the two has the lower energy in a multi-electron atom.

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

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:

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:

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) …