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APEX CLASS ESTD 2018

IITJEE CHEMISTRY: Atomic Structure & Quantum Numbers Visualizer

By

Ashish Kumar Bahekar

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APEX CLASS • IITJEE Physical Chemistry Lab

Atomic Structure & Quantum Numbers Visualizer

Master Bohr Orbits, Spectral Series, Probability Nodes (n-l-1), and Aufbau Exceptions!

🔬 Valid for single-electron species (H, He⁺, Li²⁺). Select Concept:
+Ze e⁻ (v_n) n=1 n=2 n=3 Bohr’s Orbit Proportionality Rules 1. Radius (rn): rn = 0.529 × (n² / Z) [Å] Distance between orbits increases! (r₂-r₁ < r₃-r₂) 2. Velocity (vn): vn = 2.18×10⁶ × (Z / n) [m/s] Electron slows down in higher orbits. 3. Energy (En): En = -13.6 × (Z² / n²) [eV] Total Energy = -Kinetic Energy = ½ Potential Energy 🎯 Orbit Gap Trap: As ‘n’ increases, the RADIUS gap gets LARGER, but the ENERGY gap gets SMALLER! Hydrogen Emission Spectrum (Energy drops n₂ → n₁) n=1 n=2 n=3 n=4 n=5 Lyman (UV) Balmer (Visible) Paschen (IR) Rydberg Equation & Lines 1 / λ = R_H Z² [ 1/n₁² – 1/n₂² ] • First Line (Longest λ, Min E): n₂ = n₁ + 1 • Series Limit (Shortest λ, Max E): n₂ = ∞ • Max Lines Produced = Δn(Δn + 1) / 2 Einstein’s Photoelectric Equation: E = Φ + K_max K_max Freq (ν) → ν₀ (Threshold) Slope = h -Φ (Work Fn) Incident Energy = Work Function + Max Kinetic Energy hν = hν₀ + ½ m v²_max • Intensity of Light changes Number of e⁻ ejected (Current). • Frequency of Light changes Speed/KE of e⁻ (Stopping Pot). • If ν < ν₀: NO electrons are ejected, regardless of Intensity! Standing Electron Wave (n=4) Wave-Particle Duality & Uncertainty 1. De Broglie Wavelength (λ): λ = h / p = h / mv = h / √(2mK) = h / √(2mqV) For electron: λ = √(150/V) Å ≈ 12.27 / √V Å 2. Bohr’s Quantization Derived: 2πr = nλ ⇒ mvr = n(h/2π) 3. Heisenberg: Δx · Δp ≥ h / 4π (or Δx · Δv ≥ h / 4πm)
Energy Level Formula
En = -13.6 (Z² / n²) eV
Ionization Energy = +13.6 eV (for H)
Speed of Electron
vn = c / 137 × (Z / n)
c = speed of light (3×10⁸ m/s)
Number of Spectral Lines
N = Δn(Δn + 1) / 2
Dropping from n₂ to n₁
Stopping Potential (V₀)
e V₀ = Kmax = hν – Φ
Required to stop fastest photoelectrons
🎲 Select a Quantum Mechanical Concept:
Hierarchy of Quantum Numbers (Electron’s Address) Principal (n) Shell (Size/Energy) n = 1, 2, 3… Azimuthal (l) Subshell (Shape) l = 0 to (n-1) Magnetic (m_l) Orbital (Orientation) m_l = -l to +l Spin (m_s) Electron Spin +½ (↑) or -½ (↓) Max e⁻ in Shell = 2n² s=0, p=1, d=2, f=3 Total Orbitals = 2l+1 Pauli Exclusion Pauli’s Principle: No two electrons in an atom can have the same 4 quantum numbers! s-Orbital (l=0) Spherical (Non-Directional) p-Orbital (l=1, e.g. p_x) Dumbbell (Directional) 1 Nodal Plane (YZ plane for p_x) d-Orbital (l=2, d_z²) Double Dumbbell / Donut 4πr²R² Distance (r) → 4s Orbital 3 Nodes (Where Prob = 0) 🏆 The Golden Node Formulas 1. Radial / Spherical Nodes = n – l – 1 Regions of zero probability distance from nucleus. 2. Angular Nodes / Planes = l Planes of zero probability (e.g. p_x has 1 node: YZ plane). 3. Total Nodes = n – 1 Electronic Configuration Exceptions: Chromium (Cr) & Copper (Cu) Cr (Z=24) Expected: 4s² 3d⁴ Actual: [Ar] 4s¹ 3d⁵ (Half-Filled!) ↑ 4s¹ ↑ ↑ ↑ ↑ ↑ 3d⁵ (Extra Exchange Energy) Cu (Z=29) Expected: 4s² 3d⁹ Actual: [Ar] 4s¹ 3d¹⁰ (Fully-Filled!) ↑ 4s¹ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ 3d¹⁰ (Symmetrical Stability) ⚡ Why? 1. Symmetrical distribution of electron cloud. 2. Maximum Exchange Energy (more parallel spins = more exchanges = lower energy)!
Orbital Angular Momentum
L = √(l(l+1)) × h/2π
For s-orbital (l=0) = 0!
Spin Angular Momentum
S = √(s(s+1)) × h/2π
Where s = 1/2 for a single electron
Magnetic Moment (μ)
μ = √(n(n+2)) BM
n = number of unpaired electrons
Aufbau Principle (n+l rule)
Lower (n+l) fills first
If tied, lower n fills first (4s before 3d)
🔥 Two High-Yield JEE Traps:
1. Heisenberg Limit Trap: The formula is $\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$. If given in terms of velocity: $\Delta x \cdot \Delta v \ge \frac{h}{4\pi m}$. This means a heavier particle (macroscopic) has virtually zero uncertainty!
2. Node Counting Shortcut: Total Nodes = $n – 1$. Example: For a 4p orbital (n=4, l=1), Radial Nodes = 4 – 1 – 1 = 2, Angular Nodes = l = 1. Total = 3.
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