Structure of the Atom | IISER Smart Prep

1. Core Concept

The model of the atom has evolved from solid spheres (Dalton) to the modern Quantum Mechanical Model, which views electrons as wave-like entities rather than particles confined to fixed orbits.

2. The Sub-atomic Players

Electron (e^-): Discovered by J.J. Thomson (Cathode Ray Tube). Charge-to-mass ratio (e/m) found by Thomson. R.A. Millikan (Oil Drop Experiment) determined the charge e = 1.6 × 10^-19 C.
Proton (p⁺): Discovered by Goldstein (Anode rays).
Neutron (n⁰): Discovered by Chadwick (Bombarding Beryllium with α-particles).

3. Early Atomic Models

Thomson:

"Plum Pudding"—A positive sphere with electrons embedded like plums in pudding. Failure: Could not explain alpha particle scattering.

Rutherford:

"Nuclear Model"—A dense nucleus at the center, with electrons in random orbits. Failure: According to electromagnetic theory, an accelerating electron should radiate energy and spiral into the nucleus.

4. Developments leading to Bohr's Model

Dual Nature of EMR

Electromagnetic Radiation behaves as both wave (interference, diffraction) and particle (Photoelectric effect, Black body radiation).

E = hν = hc / λ

Planck's Quantum Theory: Energy is absorbed or emitted in discrete packets called 'quanta'. (h = 6.626 × 10⁻³⁴ Js).

Photoelectric Effect:

Ejection of electrons when light of frequency > ν₀ (threshold frequency) hits a metal surface. Kinetic energy depends on frequency, not intensity.

5. Bohr’s Model for Hydrogen Atom

Postulate: Electrons move in fixed circular "stationary" orbits.
Quantization: Angular momentum is quantized: mvr = nh / 2π.
Transitions: ΔE = E_f - E_i = hν.

r_n = 0.529 × (n² / Z) Å

Radius: For Hydrogen (Z=1), a₀ = 0.529 Å (Bohr radius).

E_n = -2.18 × 10⁻¹⁸ (Z² / n²) J = -13.6 (Z² / n²) eV

Energy: Negative sign indicates the electron is bound to the nucleus.

Line Spectra of Hydrogen

Series	n₁ (Lower)	n₂ (Upper)	Spectral Region
Lyman	1	2, 3...	Ultra-violet
Balmer	2	3, 4...	Visible
Paschen	3	4, 5...	Infrared
Brackett	4	5, 6...	Infrared
Pfund	5	6, 7...	Infrared

Rydberg Formula: 1 / λ = R_H [1/n₁² - 1/n₂²] Z² (R_H ≈ 109,677 cm⁻¹)

6. Towards the Quantum Model

λ = h / mv

de Broglie (Matter Waves): Everything behaves as a wave. Explains quantization: Circumference = nλ (2πr = nλ).

Δx · Δp ≥ h / 4π

Heisenberg Uncertainty Principle: Impossible to define exact position (x) and momentum (p) simultaneously.

Result: "Orbits" are replaced by Orbitals—3D regions of space where the probability of finding an electron is high, derived from the Schrödinger Equation (Ĥψ = Eψ). ψ² represents the probability density.

7. Quantum Numbers ("Address" of an electron)

Principal (n = 1, 2, 3…): Defines shell size and energy.
Azimuthal (l = 0 to n-1): Defines orbital shape (0=s, 1=p, 2=d, 3=f).
Magnetic (m_l = -l to +l): Defines spatial orientation (e.g., p_x, p_y, p_z).
Spin (m_s = ±1/2): Direction of electron spin.

Node Calculation Table

Type	Formula	Example (3p)
Angular Nodes	`l`	1
Radial Nodes	`n - l - 1`	3 - 1 - 1 = 1
Total Nodes	`n - 1`	2

Rules for Filling Orbitals:

Aufbau Principle: Electrons occupy lowest energy orbitals first (n+l rule).
Pauli Exclusion: No two electrons can have the same four quantum numbers.
Hund's Rule: Degenerate orbitals are singly occupied with parallel spins before pairing starts.

8. Common Mistakes

Orbit vs Orbital: Bohr = Orbit (fixed path). Quantum Mechanics = Orbital (probability space).
Aufbau Trap: Fill orbitals in order of the (n+l) energy rule. If (n+l) values are identical, fill the one with the lower n first.

9. IAT Exam Focus Points

Key Areas:

Quantum Number Validity: Always check that l < n and |m_l| ≤ l.
Node Calculation: Very high yield. Memorize n - l - 1 (radial) and l (angular).
Bohr Proportionalities: Master r_n ∝ n² / Z and E_n ∝ -Z² / n². Comparing these for H, He⁺, Li²⁺ is standard.
Shapes: s = spherical, p = dumbbell, d = double-dumbbell (except d_z²).

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Structure of the Atom