Quantum Theory of the Atom: Study Notes (Ch. 8)

Quantum Theory of the Atom

Introduction

The quantum theory of the atom revolutionized our understanding of atomic structure and behavior. This chapter explores the nature of light, the development of atomic models, and the quantum mechanical principles that describe electrons in atoms.

Properties of Light

Wave-Particle Duality

• Light exhibits both wave-like and particle-like properties.

• Wavelength (\lambda): The distance between successive crests of a wave.

• Frequency (\nu): The number of wave cycles that pass a point per second.

• Speed of Light (c): All electromagnetic waves travel at $c = 3.00 \times 10^8$ m/s in a vacuum.

Equation:

$c = \lambda \nu$

• Photon: A quantum of electromagnetic radiation, carrying energy $E = h\nu$ where $h$ is Planck's constant ($6.626 \times 10^{-34}$ J·s).

Electromagnetic Spectrum

Classification of Electromagnetic Radiation

• The electromagnetic spectrum includes all types of electromagnetic radiation, from gamma rays to radio waves.

• Visible light is a small portion of the spectrum, with wavelengths from about 400 nm (violet) to 700 nm (red).

The Atomic Spectrum of Hydrogen

Line Spectra and Energy Levels

• When hydrogen gas is excited, it emits light at specific wavelengths, producing a line spectrum.

• This observation suggests that electrons in atoms occupy discrete energy levels.

Equation (Rydberg Formula for Hydrogen):

$\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$

• Where $R$ is the Rydberg constant ($1.097 \times 10^7$ m$^{-1}$), $n_1$ and $n_2$ are integers with $n_2 > n_1$.

The Bohr Model

Quantized Orbits

• Niels Bohr proposed that electrons move in circular orbits around the nucleus with quantized energies.

• Electrons can only occupy certain allowed orbits; energy is absorbed or emitted when electrons transition between these orbits.

Equation (Energy Levels of Hydrogen):

$E_n = -\frac{2.18 \times 10^{-18}\ \text{J}}{n^2}$

• Where $n$ is the principal quantum number (1, 2, 3, ...).

The Quantum Mechanical Model of the Atom

Electron Probability and Orbitals

• The quantum mechanical model describes electrons as wavefunctions, not fixed orbits.

• Orbitals are regions of space where the probability of finding an electron is high.

• The Schrödinger equation mathematically describes these wavefunctions.

Equation (Schrödinger Equation):

$\hat{H}\Psi = E\Psi$

• Where $\hat{H}$ is the Hamiltonian operator, $\Psi$ is the wavefunction, and $E$ is the energy.

Quantum Numbers

Describing Electron States

• Each electron in an atom is described by a set of four quantum numbers:

  • Principal quantum number (n): Energy level (n = 1, 2, 3, ...)

  • Angular momentum quantum number (l): Shape of the orbital (l = 0 to n-1)

  • Magnetic quantum number (m_l): Orientation of the orbital (m_l = -l to +l)

  • Spin quantum number (m_s): Electron spin (+1/2 or -1/2)

Orbital Shapes and Energies

s, p, d, and f Orbitals

• s orbitals are spherical, p orbitals are dumbbell-shaped, d and f orbitals have more complex shapes.

• Energy increases with both n and l values.

Photoelectric Effect

Evidence for Particle Nature of Light

• When light of sufficient frequency strikes a metal surface, electrons are ejected—this is the photoelectric effect.

• Demonstrates that light energy is quantized in photons.

Equation (Photoelectric Effect):

$E_{\text{photon}} = h\nu = \phi + KE_{\text{electron}}$

• Where $\phi$ is the work function (minimum energy to remove an electron), $KE$ is the kinetic energy of the ejected electron.