Quantum Mechanics & Atomic Structure

Posted on Nov 13, 2024 in Physics

Applied Quantum Mechanics at the Atomic Level

Limitations of the Bohr Model

While successful, the Bohr model was eventually superseded by quantum mechanics (wave mechanics). It couldn’t explain why electron orbits had specific energies or the periodicity of element properties. Experimental findings also challenged the model:

  • Improved spectrographs revealed that some spectral lines were doublets.
  • Spectral lines split when substances were subjected to magnetic fields.

Quantum-Mechanical Model of the Atom

Quantum mechanics provided a more accurate model. Key features include:

  • Wave-Particle Duality: In 1924, Louis de Broglie proposed that particles have wave-like properties. These matter waves are detectable in subatomic particles.
  • Heisenberg’s Uncertainty Principle (1927): There’s a limit to the precision with which a particle’s position and momentum can be simultaneously determined. This limit is negligible at the macroscopic level.

The quantum-mechanical model describes electron behavior within the atom, acknowledging their wave nature and the impossibility of predicting exact trajectories. This introduces the concept of orbitals, probability distributions of finding an electron with a given energy.

Quantum Numbers

Solving quantum-mechanical equations yields quantum numbers, which describe electron behavior:

  • Principal Quantum Number (n): Represents energy level (1, 2, 3…). Higher numbers indicate higher energy and greater distance from the nucleus.
  • Orbital Angular Momentum Quantum Number (l): Determines orbital shape and energy within each level (0 to n-1).
  • Magnetic Quantum Number (ml): Describes orbital orientation in space and explains spectral line splitting in magnetic fields.
  • Electron Spin Quantum Number (ms): Represents an intrinsic electron property (spin). Indicates alignment (parallel or antiparallel) to an external magnetic field (+1/2 or -1/2). Electrons with the same ml are said to have parallel spins or be unpaired.

Atomic Emission Spectra

All substances interact with energy. An emission spectrum is the radiation emitted by an element in gaseous form when energized. Sunlight’s spectrum is continuous, containing all frequencies. In contrast, elemental emission spectra are discontinuous, with elements emitting radiation only at specific frequencies. This unique spectrum serves as an element’s fingerprint.

Planck’s Quantum Theory

Classical electromagnetism suggests wave energy depends solely on amplitude. However, this contradicted experimental observations of radiation from heated bodies. In 1900, Max Planck proposed quantum theory: bodies emit or absorb energy in discrete packets (quanta).

Photoelectric Effect

In 1887, Heinrich Hertz discovered that electromagnetic radiation striking a metal surface ejects electrons (the photoelectric effect). Key observations:

  • Occurs when radiation frequency exceeds a metal-specific threshold (cutoff frequency, v0).
  • Emitted electron kinetic energy increases with radiation frequency.
  • Increased radiation intensity increases the number of emitted electrons, not their energy.

In 1905, Albert Einstein explained this using quantum theory. He proposed that electromagnetic radiation consists of energy quanta called photons. A photon (frequency v, energy hv) transfers its energy to an electron, which uses part to escape the metal and the rest to gain kinetic energy.

Limitations of Rutherford’s Model

Rutherford’s model, while groundbreaking, had limitations:

  • Accelerating electrons should continuously emit energy, spiraling into the nucleus. This contradicts the observed discontinuous spectra.
  • The model predicted continuous radiation, conflicting with the discrete lines observed in atomic spectra.

Bohr’s Model

Niels Bohr built upon Rutherford’s model, incorporating energy quantization to explain the hydrogen atom’s discontinuous spectrum. His 1913 model postulated:

  • Quantized electron energy within the atom (stationary states).
  • Electrons move in circular orbits corresponding to allowed energy levels (n = 1, 2, 3…).
  • Allowed energy levels have quantized angular momentum (mvr = n(h/2π), where h is Planck’s constant).
  • Energy is absorbed/emitted only during electron transitions between energy levels.