Chapter 3: Electronic Structure of the Atom

3.1 Introduction: The Transition Away from Classical Mechanics

Rutherford's gold foil experiment: Led to the nuclear model of the atom.
Classical mechanics:
- Explained physical properties of matter on the macroscopic level.
- Could not explain the properties of matter at the atomic level (e.g., what prevents electrons from spiraling inward toward the nucleus).
Quantum mechanics: Attempts to describe matter at the atomic level.

3.2 Electromagnetic Radiation

James Maxwell: Proposed that light is a form of electromagnetic radiation.
Radiation: A form of energy that can be thought of as an oscillating wave moving through space.
Characteristics of electromagnetic waves:
- Amplitude: Vertical distance from the peak or trough of a wave to its midline.
- Intensity: The square of the amplitude, related to brightness.
- Wavelength, \lambda: Peak-to-peak or trough-to-trough distance.
  - Measured in meters (m) and nanometers (nm).
  - 1 \text{ m} = 1 \times 10^9 \text{ nm}.
- Frequency, \nu: How often a complete wave cycle passes a fixed point in space.
  - 1 cycle per second = 1 Hertz (Hz).
- Relationship between wavelength and frequency:
  - A wave with a long wavelength has a small frequency.
  - A wave with a short wavelength has a high frequency.

3.2.1 The Speed of Light

Electromagnetic waves travel through a vacuum at a constant speed, denoted as c.
Speed of light, c = 2.998 \times 10^8 \text{ m/s}.
Equation relating speed, wavelength, and frequency: c = \lambda \nu
- This implies that wavelength (\lambda) and frequency (\nu) are inversely proportional.

3.2.2 Diffraction and Interference

Diffraction: When waves pass through closely spaced slits, they spread out in concentric circles, becoming new wave fronts.
Interference: As wave fronts travel, they collide or interfere with each other, creating a diffraction (or interference) pattern.
Diffraction pattern: Consists of bands of alternating intensity (bright and dark areas).
- Constructive interference: Occurs when waves combine to produce a wave with a larger amplitude (bright bands).
- Destructive interference: Occurs when waves combine to cancel each other out, resulting in zero amplitude (dark bands).

3.3 The Electromagnetic Spectrum

Different types of radiation comprise the electromagnetic spectrum.
Energy-frequency relationship: E = h \nu
- Where h is Planck's constant, h = 6.626 \times 10^{-34} \text{ J} \cdot \text{s}.
Energy-wavelength relationship: E = \frac{hc}{\lambda}.

3.4 Challenges to Classical Physics

3.4.1 Atomic Spectroscopy

Different elements emit specific colors of light (e.g., sodium vapor lamps emit yellow light; hydrogen emits an intense blue-purple light in a gas discharge tube).
When emitted light from hydrogen gas is directed to a prism, it is separated into its component wavelengths or frequencies, creating a photoemission spectrum.
- Continuous spectrum: Contains all wavelengths (like white light through a prism).
- Line spectrum: Shows only specific, discrete wavelengths (characteristic of excited atoms).

3.4.1 The Rydberg Equation for Hydrogen

The Rydberg equation gives all wavelengths of radiation emitted when hydrogen (H) is excited.
Rydberg Equation (wavelength): \frac{1}{\lambda} = R \left(\frac{1}{nf^2} - \frac{1}{ni^2}\right)
Rydberg Equation (frequency): \nu = R \left(\frac{1}{nf^2} - \frac{1}{ni^2}\right)
- Where:
  - R is the Rydberg constant, which can be expressed as 1.097 \times 10^7 \text{ m}^{-1} (for wavelength calculation) or 3.29 \times 10^{15} \text{ Hz} (for frequency calculation).
  - \lambda is the wavelength of the electromagnetic wave in a vacuum.
  - \nu is the frequency of the electromagnetic wave.
  - nf and ni are integers (\ge 1) representing the final and initial energy levels, respectively, such that nf < ni.

3.4.2 Blackbody Radiation

Classical mechanics view: A blackbody is an idealized object capable of absorbing and emitting all frequencies of radiation.
- As a blackbody is heated, electrons oscillate, releasing energy as electromagnetic radiation.
- Rayleigh-Jeans law: A classical physics prediction that failed to accurately describe blackbody radiation at all frequencies.
- Ultraviolet catastrophe: The specific failure of the Rayleigh-Jeans law to predict blackbody radiation at high frequencies (short wavelengths).
Max Planck (1900): Broke from the classical view by proposing that energy is absorbed and emitted in the form of discrete, quantized packets called quanta.
- Planck's equation: E = h \nu or E = \frac{hc}{\lambda}.
  - Where h is Planck's constant ( 6.626 \times 10^{-34} \text{ J} \cdot \text{s} ).

3.4.3 Hertz, Einstein, and the Photoelectric Effect

Heinrich Hertz (1886-1887): Observed the photoelectric effect, where a metal, upon absorbing sufficient energy, ejects electrons with measurable velocity.
- Threshold frequency: The minimum frequency of radiation required to eject electrons, dependent on the identity of the metal but independent of the intensity of the radiation.
- Higher intensity radiation results in more electrons being ejected, but only if the frequency is at or above the threshold frequency.
Albert Einstein (1905): Expanded upon Planck's work by characterizing light as discrete particles called photons.
- Each photon has an energy E = h \nu.
- The kinetic energy (KE) of the ejected electron equals the amount of energy in excess of the threshold energy (work function), \phi.
- The electronic energy in the atom is quantized.
- Photoelectric effect equation: KE = \frac{1}{2} m v^2 = h \nu - \phi

3.4.4 Photoemission Revisited: The Bohr Model of the Atom

Niels Bohr (1914): Proposed a model where electrons traveled in circular paths around the nucleus in stationary orbits.
Atomic photoemission explanation:
- An electron absorbs incident energy, becomes excited, and is promoted to a higher energy state.
- The excited electron then relaxes, returning to a state of lower energy.
- During relaxation, the electron releases some of its energy in the form of electromagnetic radiation (a photon).
Energy of an electron at an orbit n (for hydrogen): E_n = -\frac{R}{n^2}
- Where R = 2.18 \times 10^{-18} \text{ J} is the Rydberg constant (in energy units).
- n is the principal quantum number.
- n = 1 is the lowest possible energy state, called the ground state.
- All states where n > 1 are excited states. As n increases, the excited state becomes higher in energy, and the electron is further away from the nucleus.
Energy of an emitted photon (Bohr frequency condition): \Delta En = -R \left(\frac{1}{nf^2} - \frac{1}{n_i^2}\right)
- This equation explains the discrete lines in the hydrogen emission spectrum.

3.4.5 The de Broglie Wavelength and the Wave-Particle Duality of Matter

Louis de Broglie (1924): Suggested that matter possesses a wave-particle duality, similar to light.
- All matter, especially subatomic particles, could be treated as a moving wave.
- This wave nature is relatively significant for small subatomic particles.
de Broglie wavelength equation: \lambda = \frac{h}{p} = \frac{h}{mv}
- Where h is Planck's constant.
- p is the linear momentum.
- m is mass in kilograms (kg).
- v is velocity in meters per second (m/s).
Diffraction Patterns: G.P. Thompson and Davisson/Germer (1924) demonstrated that electrons (matter) produce diffraction patterns similar to those of X-rays (light), providing experimental evidence for wave-particle duality in matter.

3.4.6 The Heisenberg Uncertainty Principle

Werner Heisenberg: Proposed that nothing could simultaneously be described as both a wave and a particle with absolute certainty.
Heisenberg Uncertainty Principle: It is impossible to describe with absolute certainty both the exact position (particle character, \Delta x) and momentum (wave character, \Delta p) of an electron.
- Equation: \Delta x \Delta p \ge \frac{h}{2\pi}
- Where \Delta x is the uncertainty in position and \Delta p = m \Delta v is the uncertainty in momentum (velocity).
The Bohr model, with its quantized orbits and energy levels, was successful for the hydrogen atom but failed for polyelectronic atoms due to this inherent uncertainty.

3.5 Schrödinger, Born, and Atomic Orbital Wavefunctions

Schrödinger: Successfully calculated the electron energy levels of the hydrogen atom using his equation.
Schrödinger Equation: H \Psi = E \Psi
- H is the Hamiltonian operator.
- \Psi (psi) is a wavefunction, a mathematical description of a quantum particle.
- E is the total energy.
While the wavefunction \Psi itself does not have direct physical meaning, |\Psi|^2 is the probability density of the electron.
- |\Psi|^2 tells us where the electron is likely to be found around an atom.

3.5.1 Radial Wavefunctions & 3.5.2 Radial Distribution Plots

An electron’s wavefunction \Psi can be separated into radial and angular components.
The radial probability density accounts for the increasing volume of each layer at a greater distance from the nucleus.

3.5.2 Angular Wavefunctions

(No specific details provided in the transcript beyond its existence as a component).

3.6 Quantum Numbers

Quantum numbers: Quantize certain properties of the electron, having specific allowed values. They arise from Schrödinger's equation and lead to the periodic table structure.

3.6.1 The Principal Quantum Number, n

Symbol: n
Physical meaning: Quantizes the energy of an electron and determines the most likely distance an electron is away from the nucleus.
Allowed values: n = 1, 2, 3, 4, \dots
Designation: Refers to the shell an electron occupies.

3.6.2 The Orbital Angular Momentum Quantum Number, \ell

Symbol: \ell
Physical meaning: Quantizes the angular momentum of an electron and determines the shape of the orbital.
Allowed values: \ell = 0, 1, 2, 3, \dots, n - 1
Designation: Refers to subshells.
spdf notation:
- \ell = 0 \implies s orbital.
- \ell = 1 \implies p orbital.
- \ell = 2 \implies d orbital.
- \ell = 3 \implies f orbital.
An orbital described by quantum numbers n = 2 and \ell = 1 is referred to as a 2p orbital.

3.6.3 Shapes of Atomic Orbitals

s-type orbitals: Radial wavefunctions do not change over all angles, leading to spherically shaped orbitals.
Radial nodes: The number of radial nodes in an orbital is given by n - \ell - 1.

3.6.4 Magnetic Quantum Number, m_{\ell}

Symbol: m_{\ell}
Physical meaning: Gives orbitals their specific orientations in space.
Allowed values: m_{\ell} = -\ell, -\ell+1, \dots, 0, \dots, \ell-1, \ell
Angular nodes: These are flat planes between lobes of orbitals (e.g., p and d orbitals).

3.6.5 The Spin Quantum Number, m_s

Symbol: m_s
Physical meaning: Describes the intrinsic angular momentum (spin) of an electron, which behaves like a tiny magnet.
Allowed values:
- m_s = +\frac{1}{2} (spin up,