Lecture Notes on Electron Structure and Quantum Mechanics

Key Concepts

Energy Levels and Orbitals

• Hydrogen atom energy levels are quantized.
• Energy of an nth orbital for hydrogen:
  • Formula:
    [ E_n = -\frac{13.6}{n^2} \text{ eV} ]
• Radius of nth orbital:
  • Formula:
    [ r_n = a_0 \cdot n^2 ]
  • Where:
    • ( a_0 = 0.529 \text{ Å} )
• Velocity of electron in nth orbital:
  • Formula:
    [ V_n = V_0 \cdot \frac{Z}{n} ]
  • Where:
    • ( V_0 = 2.188 \times 10^8 \text{ cm/s} )

Quantum Mechanics

• Angular momentum is quantized:
  • Formula:
    [ L = mvr = n \cdot \frac{h}{2\pi} ]
• Energy difference between levels:
  • ( E_{2} - E_{1} ) results in spectral lines.

Limitations of the Model

• Fails to explain multi-electron systems.
• Cannot account for electron-electron interactions in multi-electron atoms.

Important Formulas

Rydberg Formula

• Energy transitions in hydrogen:
  • [ E = hc(\bar{\nu}) = \frac{hc}{\lambda} ]
• Wave number relation:
  • ( \bar{\nu} = \frac{1}{\lambda} )

De Broglie Wavelength

• Wave-particle duality:
  • Formula:
    [ \lambda = \frac{h}{mv} ]
• Momentum (p):
  • ( p = mv )

Uncertainty Principle

• Heisenberg's uncertainty principle:
  • Position and momentum cannot be known precisely at the same time.

Quantum Numbers

• Principal quantum number (n): Energy level
• Azimuthal quantum number (l): Orbital shape
• Magnetic quantum number (m): Orientation of orbital

Electron Configurations

• Aufbau principle: Electrons fill the lowest energy orbitals first.
• Pauli exclusion principle: No two electrons can have the same set of quantum numbers.
• Hund's rule: Electrons will fill degenerate orbitals singly before pairing.

Orbital Shapes and Nodes

• Orbital shapes:
  • s: spherical, p: dumbbell, d: complex shapes
• Nodes: Regions where the probability of finding an electron is zero.
• Radial nodes:
  • Formula:
    [ n - l - 1 ]
• Total nodes:
  • Formula:
    [ n - 1 ]

Summary

• Understanding the quantum mechanical model is crucial for explaining atomic structure and electron behavior.
• Key topics include energy levels, quantum numbers, orbital shapes, and electron configurations.

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Homework Assignments

• Calculate the radius of the third orbit for a hydrogen atom.
• Explore the concepts of dual nature of matter further.