Stability of Fully Filled Subshells
The stability of fully filled subshells in atoms is a key concept in understanding atomic structure and electron configurations. The main idea revolves around the arrangement of electrons within an atom's orbitals and how this configuration impacts the atom's overall stability.
Quantum Mechanics and Electron Configuration
In quantum mechanics, electrons occupy specific energy levels known as orbitals. These orbitals are grouped into subshells denoted by the letters $s, p, d,$ and $f$. The stability of an atom greatly depends on how these orbitals are filled.
Pauli Exclusion Principle and Hund's Rule

Pauli Exclusion Principle: This principle states that no two electrons in an atom can have the same set of quantum numbers. In simple terms, an orbital can hold a maximum of two electrons with opposite spins.

Hund's Rule: According to Hund's rule, electrons will first occupy different orbitals within the same subshell singly and with parallel spins before pairing up.
Increased Stability of Fully Filled Subshells
Atoms achieve enhanced stability when their subshells are either completely filled or halffilled. This increased stability is due to:
 Symmetrical Distribution: Fully filled or halffilled subshells result in a more symmetrical and balanced distribution of electrons, leading to lower energy states.
 Exchange Energy: Full and halffilled configurations maximize the exchange energy, which is a stabilizing factor associated with parallel electron spins.
For instance, the electron configuration of noble gases, such as neon ($Ne$) and argon ($Ar$), demonstrates full $p$ subshells:
Neon ($Ne$): $1s^2\,2s^2\,2p^6$
Argon ($Ar$): $1s^2\,2s^2\,2p^6\,3s^2\,3p^6$
These configurations show full $s$ and $p$ subshells, making these atoms extremely stable and chemically inert.
Mathematical Representation
The stability can be mathematically understood by considering the exchange integral $K$, which represents the exchange energy due to symmetric wavefunctions of parallel spins. For instance, for a fully filled $p$subshell:
$\Delta E_{exchange} = K \sum_{i<j} \langle \phi_i \phi_j  \frac{1}{r_{12}}  \phi_i \phi_j \rangle$
Here, $\phi$ represents the wavefunctions, and $r_{12}$ is the distance between two electrons. Fully filled subshells maximize this stabilizing exchange interaction.
RealWorld Examples
 Chromium ($Cr$): Despite violating the Aufbau principle, chromium has a $3d^5\,4s^1$ configuration instead of $3d^4\,4s^2$. This is because a halffilled $3d$subshell provides extra stability.
 Copper ($Cu$): Copper exhibits a $3d^{10}\,4s^1$ configuration rather than $3d^9\,4s^2$, favoring the fully filled $d$subshell for enhanced stability.
In summary, the stability of fully filled subshells is a fundamental concept that explains the preferred electron configurations of atoms and their resulting stabilities. This understanding is crucial for predicting and explaining various chemical properties and reactivities.