Unit 1: Atomic Structure

Review of Early Models
Hydrogen Atom Spectra
Schrödinger Wave Equation
Quantum Numbers
Electron Filling and Configuration

Review of Early Models

Bohr's Theory and its Limitations

Niels Bohr proposed a model for the hydrogen atom where electrons revolve in fixed, circular "orbits" with quantized angular momentum (mvr = n(h) / (2π)).

Limitations of Bohr's Theory:

Fails for Multi-electron Atoms: It could not explain the spectra of atoms with more than one electron.
Zeeman and Stark Effects: It could not explain the splitting of spectral lines in a magnetic field (Zeeman) or an electric field (Stark).
Violates Heisenberg's Principle: It defines a fixed path (orbit) and velocity, which contradicts the Uncertainty Principle.
No 3D Model: It did not explain the 3D nature of the atom or the shapes of orbitals.

Dual Behaviour of Matter (de Broglie's Relation)

Louis de Broglie proposed that all matter (like electrons) exhibits wave-particle duality. A particle with momentum p has an associated wavelength λ.

Formula (de Broglie Relation): λ = (h) / (p) = (h) / (mv)
Where: h = Planck's constant, m = mass, v = velocity

Heisenberg Uncertainty Principle

This principle states that it is impossible to simultaneously measure or know both the exact position (Δ x) and the exact momentum (Δ p) of a microscopic particle.

Formula: Δ x · Δ p ≥ (h) / (4π)

Significance: This principle refutes Bohr's idea of fixed orbits and introduces the concept of probability and orbitals. This led to the "Need of a new approach to Atomic structure."

Hydrogen Atom Spectra

When an electron in an excited state (higher n) drops to a lower state (lower n), it emits a photon, creating a spectral line. The wavelength of this line is given by the Rydberg formula:

Formula: (1) / (λ) = R_H ( (1) / (n₁²) - (1) / (n₂²) )
Where: R_H = Rydberg constant (109,677 cm^-1), n₁ = lower energy level, n₂ = higher energy level

Spectral Series for Hydrogen
Series Name	n₁ (Final)	n₂ (Initial)	Region of Spectrum
Lyman	1	2, 3, 4, ...	Ultraviolet (UV)
Balmer	2	3, 4, 5, ...	Visible
Paschen	3	4, 5, 6, ...	Infrared (IR)

Schrödinger Wave Equation

The Time independent Schrödinger equation is the fundamental equation of quantum mechanics. It describes the electron as a wave, and its solutions (wave functions) define the allowed energy states and shapes of orbitals.

Equation: Ĥψ = Eψ

Meaning of Various Terms in it:

Ĥ (Hamiltonian Operator): A mathematical operator that represents the total energy (Kinetic + Potential) of the system.
ψ (Wave Function): The solution to the equation. It is an "amplitude function" and has no direct physical meaning itself.
E (Eigenvalue): A specific, allowed energy value for the system (e.g., the energy of the 1s orbital).
ψ² (Probability Density): The square of the wave function (ψ²) at any point in space gives the probability of finding the electron at that point. An orbital is a 3D region where this probability is high (>90%).

Quantum Numbers and their Significance

The solutions to the Schrödinger equation are characterized by three quantum numbers (n, l, m). A fourth (s) was added to describe the electron itself.

The Four Quantum Numbers
Quantum Number	Symbol	Allowed Values	Significance
Principal	n	1, 2, 3, ... (positive integers)	Determines the main energy level (shell) and size of the orbital.
Azimuthal (Angular Momentum)	l	0 to (n-1)	Determines the subshell (s, p, d, f) and the shape of the orbital. (l=0 is s, l=1 is p, l=2 is d)
Magnetic	m_l	-l to 0 to +l	Determines the orientation of the orbital in space (e.g., p_x, p_y, p_z).
Spin	m_s	+(1) / (2) or -(1) / (2)	Determines the intrinsic spin of the electron (spin up ↑ or spin down ↓).

Electron Filling and Configuration

Rules for filling electrons in various orbitals

Aufbau Principle: Electrons fill orbitals starting from the lowest available energy level. The order is given by the (n+l) rule:
- Lower (n+l) value fills first.
- If (n+l) is equal, the orbital with the lower n value fills first.
- Order: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p ...
Pauli Exclusion Principle: No two electrons in an atom can have the same set of all four quantum numbers. This implies an orbital can hold a maximum of two electrons, and they must have opposite spins (↑↓).
Hund's Rule of Maximum Multiplicity: For degenerate orbitals (e.g., p_x, p_y, p_z), pairing of electrons does not start until each orbital is occupied by one electron (half-filled). The most stable state has the maximum number of unpaired, parallel spins.

Electronic Configurations of the Atoms

This is the notation showing the distribution of electrons among orbitals.

Nitrogen (Z=7): 1s² 2s² 2p³ (Orbital: [↑↓] [↑↓] [ ↑ ][ ↑ ][ ↑ ])
Iron (Z=26): 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶ or [Ar] 4s² 3d⁶

Stability of half-filled and completely filled orbitals

Subshells that are exactly half-filled (p³, d⁵, f⁷) or completely filled (p⁶, d¹⁰, f¹⁴) have extra stability. This is due to:

Symmetrical Distribution: A symmetrical (half or full) distribution of electrons leads to greater stability.

Concept of Exchange Energy: This is a quantum mechanical effect. Electrons with the same spin (parallel) in degenerate orbitals can "exchange" their positions. Each such exchange releases energy, stabilizing the atom. The more parallel electrons, the more exchanges are possible, and the greater the stability. A d⁵ configuration has 10 possible exchanges, while a d⁴ has only 6.

Anomalous Electronic Configurations

Due to the extra stability of half-filled and filled d-subshells, the Aufbau principle is sometimes violated (Relative energies of atomic orbitals shift).

Key Exceptions to Know:

Chromium (Cr, Z=24):
- Expected: [Ar] 4s² 3d⁴
- Actual: [Ar] 4s¹ 3d⁵ (to get a stable d⁵ configuration)
Copper (Cu, Z=29):
- Expected: [Ar] 4s² 3d⁹
- Actual: [Ar] 4s¹ 3d¹⁰ (to get a stable d¹⁰ configuration)