Electron Configurations & Quantum Theory Essentials

Bohr Model

• Classical model failed: predicted electrons spiral into nucleus & continuous light emission; atoms stable & show line spectra instead.
• Bohr postulates
  • Electrons occupy fixed circular orbits with principal quantum number $n = 1,2,3,\dots$
  • Orbit radii $\propto n^2$; larger $n$ ⇒ farther from nucleus.
  • Electrons in a permitted orbit do not radiate energy.
  • Energy emitted/absorbed only during transitions between orbits.
  • Energy of level $n$ (for H-like atoms): $E<em>n = \frac{E</em>1}{n^2}$ with $E<em>1 = -13.6\,\text{eV}$. • Examples: $n=2 \Rightarrow E</em>2 = -3.4\,\text{eV}$; $n=3 \Rightarrow -1.5\,\text{eV}$; $n=4 \Rightarrow -0.85\,\text{eV}$.
• Photon energy $\Delta E = E<em>{final}-E</em>{initial}$; $10.2\,\text{eV}$ ( $n=2 \to 1$ ) corresponds to UV light.

Energy Levels & Spectra

• Ionization: $n \to \infty$, $E=0$ (electron free).
• Emission lines: electrons drop to lower $n$; photons with discrete energies.
• Absorption lines: same energy gaps absorbed when electrons promoted.

Quantum Numbers

• 4 numbers (no two electrons share all four):
  1. $n$ – principal; shell, energy, size; higher $n$ ⇒ higher (less negative) energy.
  2. $l$ – angular (
     • l = 0,1,2,3\,\dots$ → s,p,d,f); defines subshell shape.
  3. $m_l$ – magnetic; orbital orientation; values $-l \dots +l$ ( gives # of orbitals: s=1, p=3, d=5, f=7 ).
  4. $m_s = \pm \frac12$ – spin; two electrons per orbital with opposite spins.
• Wave-mechanical model (Schrödinger): electrons described by probability clouds, not fixed paths; explains multi-electron atoms.

Electron Orbital Filling Rules

• Aufbau: fill lowest-energy orbitals first (order: $1s \to 2s \to 2p \to 3s \to 3p \to 4s \to 3d \to 4p \dots$).
• Pauli Exclusion: max 2 electrons per orbital, opposite $m_s$.
• Hund: within a subshell, maximize unpaired & same-spin electrons before pairing.

Electron Configurations

• Notation: \text{shell letter}^{\text{# e⁻}} e.g., $1s^2$.
• Total superscripts = atomic number for neutral atom.
• Examples
  • Na: $1s^2 2s^2 2p^6 3s^1$
  • Kr: $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6$
• Common violations in orbital diagrams
  • Wrong spin pairing ⇒ Pauli violation.
  • Skipping lower energy subshell ⇒ Aufbau violation.
  • Pairing before all orbitals singly filled ⇒ Hund violation.

Quick Facts

• Stable atoms: negative energy levels keep electrons bound.
• As $n$ ↑, energy gaps shrink; high $n$ levels nearly continuous.
• Both electrons & light exhibit wave-particle duality.