Quantum Mechanical Model of the Atom
Why Quantum Mechanics? 🤔
Classical physics (like Newton’s laws) works for big things (planets, stones) but fails for tiny particles like electrons. Why? It ignores two key ideas:
- 🌊 Wave-particle duality (matter has both wave & particle properties)
- 🎯 Heisenberg’s uncertainty principle (can’t know exact position and velocity of an electron at once)
Quantum mechanics fixes this! It describes how electrons really behave in atoms.
Schrödinger’s Equation: The Heart ♥️
Erwin Schrödinger (1887–1961) created the fundamental equation of quantum mechanics:
\[\hat{H} \psi = E \psi\]
- \(\hat{H}\) = Hamiltonian (math operator for total energy)
- \(\psi\) = Wave function (describes electron’s behavior)
- \(E\) = Energy of the electron
Solving this equation gives possible energy levels (E) and wave functions (\(\psi\)) for electrons.
Atomic Orbitals & Quantum Numbers 🔢
An atomic orbital is defined by its wave function \(\psi\). Each orbital is labeled by 3 quantum numbers:
| Quantum Number | Symbol | What it Determines | Allowed Values |
|---|---|---|---|
| Principal | \(n\) | Size & energy of orbital (also called shells: K, L, M, N…) | 1, 2, 3, … |
| Azimuthal (Angular Momentum) | \(l\) | Shape of orbital (also called subshells: s, p, d, f…) | 0 to \(n-1\) (s=0, p=1, d=2, f=3) |
| Magnetic | \(m_l\) | Orientation in space | \(-l\) to \(+l\) |
➕ Electron Spin Quantum Number (\(m_s\)): Describes electron’s spin. Only two values: \(+\frac{1}{2}\) (↑) or \(-\frac{1}{2}\) (↓).
Shapes of Orbitals 🌀
- s-orbitals: Spherical ⚪ (size increases with \(n\))
- p-orbitals: Dumbbell-shaped 🎯 (3 orientations: px, py, pz)
- d-orbitals: Cloverleaf or unique 🍀 (5 orientations: dxy, dyz, dxz, dx²-y², dz²)
Node Alert! Regions where probability of finding electron is zero (e.g., 2s has 1 node, 3s has 2 nodes).
Orbitals vs. Orbits 🚫
Bohr orbits (circular paths) don’t exist! Electrons aren’t in fixed paths. Instead, orbitals are regions where electrons are probably found (described by \(\psi\)).
Energy of Orbitals ⚡
Hydrogen: Energy depends only on \(n\):
\(1s < 2s = 2p < 3s = 3p = 3d < ...\)
Multi-electron atoms: Energy depends on both \(n\) and \(l\)! Order changes:
\(1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < ...\)
Why? Electron repulsion & shielding effects.
Filling Electrons: 3 Golden Rules ✨
- Aufbau Principle: Fill orbitals from lowest energy first.
Order: 1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → … - Pauli Exclusion: Max 2 electrons per orbital, with opposite spins (↑↓).
- Hund’s Rule: Fill degenerate orbitals (same energy) singly first, then pair up.
Example: Carbon → 2p2 is: ↑ ↑ not ↑↓
Electronic Configuration Examples 💡
- H (1): \(1s^1\)
- He (2): \(1s^2\)
- C (6): \(1s^2 2s^2 2p^2\)
- Na (11): \(1s^2 2s^2 2p^6 3s^1\) or [Ne] 3s1
- Exceptions: Cr (24): [Ar] \(3d^5 4s^1\) (not \(3d^4 4s^2\))
Cu (29): [Ar] \(3d^{10} 4s^1\) (not \(3d^9 4s^2\))
Stability Superpowers! 🦸
Half-filled (\(d^5\), \(p^3\)) or fully-filled (\(d^{10}\), \(p^6\)) subshells are extra stable due to:
- Symmetrical electron distribution
- High exchange energy (electrons with same spin swap positions)
Top 5 NEET Must-Knows! 🚀
- Quantum Numbers: Know \(n\), \(l\), \(m_l\), \(m_s\) definitions & allowed values.
- Orbital Shapes & Nodes: Recognize s/p/d shapes; calculate nodes = \(n-1\).
- Electron Configuration: Write configurations for elements (especially Cr, Cu exceptions!).
- Aufbau Order: Memorize filling sequence (4s before 3d!).
- Hund’s Rule & Stability: Explain why half/fully filled subshells are special.
