Electron configurations Knowledge-Based Study

Study notes

1. The Architecture of the Electron Cloud

   Electrons in an atom are arranged in discrete energy levels around the nucleus. Understanding this arrangement requires familiarity with three fundamental components:

   • Principal Energy Levels (n): Designated by integers ($n = 1, 2, 3, \dots$), these levels correspond to the general distance of an electron from the nucleus and its overall energy. Higher values of $n$ indicate greater average distance and higher potential energy.
   • Subshells ($l$): Each principal energy level consists of one or more subshells, identified by the letters s, p, d, and f. The number of subshells in an energy level equals the principal quantum number $n$.
   • Orbitals: An orbital is a three-dimensional region of space where there is a high probability of finding an electron. Each individual orbital can accommodate a maximum of two electrons.

   Capacities of Subshells

   Each type of subshell contains a characteristic number of orbitals and a corresponding maximum electron capacity:

   • s subshell: 1 orbital, holds up to 2 electrons
   • p subshell: 3 orbitals, holds up to 6 electrons
   • d subshell: 5 orbitals, holds up to 10 electrons
   • f subshell: 7 orbitals, holds up to 14 electrons

   Fundamental Principles

   Electrons occupy orbitals in a systematic order that minimizes the total energy of the atom. Three core rules govern this ground-state electron distribution:

   1. The Aufbau Principle

   Electrons occupy the available orbitals of lowest energy before filling higher-energy orbitals. The relative energy of subshells is generally predicted by the $(n + l)$ rule, giving the standard filling sequence:

   $1s ightarrow 2s ightarrow 2p ightarrow 3s ightarrow 3p ightarrow 4s ightarrow 3d ightarrow 4p ightarrow 5s \dots$

   Notice that the $4s$ subshell fills before the $3d$ subshell because it is lower in energy in neutral atoms prior to filling.

   2. The Pauli Exclusion Principle

   No two electrons in an atom can have identical sets of four quantum numbers. In practical terms, an orbital can hold a maximum of two electrons, and they must have opposite spins (represented by up and down arrows: $\uparrow\downarrow$).

   3. Hund's Rule

   When degenerate orbitals (orbitals of equal energy, such as the three $2p$ orbitals) are available, electrons occupy them singly with parallel spins before any pairing occurs. This minimizes electron-electron repulsion within the subshell.

   Standard spdf Notation

   Electron configurations describe the exact distribution of electrons among available subshells. In this notation, the principal energy level is written first, followed by the subshell letter, with a superscript indicating the number of electrons residing in that subshell.

   Step-by-Step Examples

   To determine the ground-state electron configuration of a neutral atom, find its atomic number ($Z$), which equals the total number of electrons, and assign them in order of increasing subshell energy:

   • Hydrogen ($Z = 1$): Has 1 electron.
     $1s^1$
   • Carbon ($Z = 6$): Has 6 electrons.
     $1s^2\, 2s^2\, 2p^2$
   • Sodium ($Z = 11$): Has 11 electrons.
     $1s^2\, 2s^2\, 2p^6\, 3s^1$
   • Scandium ($Z = 21$): Has 21 electrons.
     $1s^2\, 2s^2\, 2p^6\, 3s^2\, 3p^6\, 4s^2\, 3d^1$

   Always verify your answer by summing the superscripts; the total must equal the atomic number of the neutral atom.

   Condensed Notation

   As atomic numbers increase, writing complete configurations becomes tedious. To simplify this, chemists use noble gas notation (condensed configuration). The symbol of the preceding noble gas is placed in brackets to represent all filled core electrons, followed by the remaining outer-shell electrons.

   Examples of Condensed Configurations

   • Magnesium ($Z = 12$): Preceded by neon ($[\text{Ne}] = 1s^2\, 2s^2\, 2p^6$).
     $[\text{Ne}]\, 3s^2$
   • Chlorine ($Z = 17$): Preceded by neon.
     $[\text{Ne}]\, 3s^2\, 3p^5$
   • Iron ($Z = 26$): Preceded by argon ($[\text{Ar}] = 1s^2\, 2s^2\, 2p^6\, 3s^2\, 3p^6$).
     $[\text{Ar}]\, 4s^2\, 3d^6$

   Core vs. Valence Electrons

   Core electrons occupy completely filled inner shells and are chemically inert. Valence electrons occupy the outermost principal energy level (and incompletely filled $d$ or $f$ subshells). Valence electrons dictate the chemical reactivity, bonding characteristics, and periodic trends of an element.

   Anomalies in the 3d Transition Series

   While the Aufbau sequence accurately predicts configurations for most elements, slight deviations occur when subshells are close in energy. Enhanced stability is associated with half-filled ($d^5$) and fully filled ($d^{10}$) subshells due to symmetrical charge distribution and maximized exchange energy.

   Chromium ($Z = 24$)

   Expected: $[\text{Ar}]\, 4s^2\, 3d^4$
   Observed: $[\text{Ar}]\, 4s^1\, 3d^5$
   Promoting one electron from $4s$ to $3d$ creates two half-filled subshells, lowering the overall energy of the atom.

   Copper ($Z = 29$)

   Expected: $[\text{Ar}]\, 4s^2\, 3d^9$
   Observed: $[\text{Ar}]\, 4s^1\, 3d^{10}$
   A completely filled $3d$ subshell provides extra thermodynamic stability over the predicted arrangement.

   Electron Removal in Transition Metal Cations

   When transition metals form positive ions, electrons are lost from the outermost principal energy level first. Therefore, electrons in the $4s$ subshell are removed before electrons in the $3d$ subshell:

   • $\text{Fe}$: $[\text{Ar}]\, 4s^2\, 3d^6$
   • $\text{Fe}^{2+}$: $[\text{Ar}]\, 3d^6$ (loses two $4s$ electrons)
   • $\text{Fe}^{3+}$: $[\text{Ar}]\, 3d^5$ (loses two $4s$ electrons and one $3d$ electron)