Quantum-Mechanical Model of the Atom
The Quantum-Mechanical Model of the Atom
Module Learning Goals
By the end of this module, students will be able to:
Explain the need for the development of the quantum mechanical model of the atom and identify key scientists who contributed to its development.
Describe the evidence for the wave/particle duality of electrons and photons.
Describe the electronic configuration of an atom or ion using the four quantum numbers and relate these to the number of nodes in an atom.
Recognize how the quantum mechanical model of the atom is reflected in the organization of the periodic table.
Use Hund’s rule and the Aufbau principle to write electron configurations for atoms and ions.
Use the Bohr Equation to calculate the energy of electronic transitions in elements.
Interpret results from the Heisenberg Equation regarding probability.
Early Views of Matter
Ancient Greeks:
Proposed metals like gold, silver, tin, lead made from elemental matter.
Matter was thought to consist of four fundamental elements: earth, water, air, and fire.
16th Century:
Rise of Alchemy, focusing on the transmutation of lead into gold.
Three main elements identified: salt, sulphur, and mercury.
1800s:
John Dalton formulated atomic theory, positing that all matter consists of small, indivisible particles.
Major Discoveries Leading to Atomic Theory
J.J. Thompson (1904):
Discovered the electron using more modern techniques and equipment.
Identified the electron as a negatively charged particle, significantly smaller than atoms.
Ernest Rutherford (1911):
Published discovery of the proton and proposed the nuclear model of the atom.
Conducted experiments using students Ernest Marsden and Hans Geiger, highlighting the nucleus of the atom.
Identified the proton as larger than the electron and positively charged.
Neutron Discovery:
Predated later discoveries, neutron mass similar to protons but no charge.
The Nuclear Model of the Atom (Pre-1927)
The atom comprises of three main particles:
Protons: +1 charge, located in the nucleus.
Neutrons: 0 charge, located in the nucleus.
Electrons: -1 charge, orbiting the nucleus.
Electrons are significantly less massive than protons and neutrons; specifically, protons and neutrons are about 2000 times more massive than electrons.
This model, while foundational, was subject to replacement due to its limitations.
The Solvay Conference, 1927
A pivotal gathering of leading scientific minds to discuss revolutionary theories that were emerging around quantum mechanics.
Light: Particle or Wave?
Poll on whether light is a particle, wave, or both yielded that 86% believe light is both.
The Nature of Light
James Clerk Maxwell (1873):
Proposed that visible light is composed of electromagnetic waves, containing oscillating magnetic and electric fields.
Wave Properties:
Can be described using amplitude, wavelength ($ ext{λ}$), and frequency ($
u$) with the equation:
u = rac{c}{ ext{λ}}, where c = 3 imes 10^8 ext{ m/s}.
Electromagnetic Radiation Spectrum
Defined as energy emitted and transmitted via electromagnetic waves.
Properties of Waves:
Interference:
Constructive interference occurs when two waves are in phase.
Destructive interference occurs when two waves are out of phase.
Diffraction:
Bending of waves around obstacles. Light passing through a small opening diffracts, confirming it behaves as a wave.
Evidence for Wave Nature of Light
Observed diffraction patterns using double-slit experiments demonstrating light's wave nature, producing light and dark bands through constructive and destructive interference.
Blackbody Radiation
Blackbody: A theoretical object absorbing all radiation and re-emitting it across a broad frequency spectrum.
Intensity of blackbody radiation varies with frequency; higher temperatures correlate with higher frequencies in emission.
The UV Catastrophe
Classical physics inadequately described blackbody radiation at short wavelengths, leading to contradictions known as the ultraviolet catastrophe.
This shortcoming prompted solutions leading to the birth of quantum mechanics.
Birth of Quantum Mechanics
Max Planck (1900):
Addressed the ultraviolet catastrophe with the concept of quantized energy levels.
Energy emitted or absorbed is only in discrete amounts termed quanta, leading to the equation:
E = h
u, where h = 6.626 imes 10^{-34} ext{ J·s}.
Planck's findings resolved the issue of the ultraviolet catastrophe and established the basis for quantum theory.
The Planck Constant and Energy Quanta
Planck's equation describes blackbody energy emission, showing the intensity correlation with frequency and resolving classical physics discrepancies.
Worked Example: Using The Planck Equation
Calculate energy of a photon of green light with wavelength 5200 Å:
Convert wavelength to meters:
ext{λ} = 5200 ext{ Å} imes 10^{-10} ext{ m}.Use speed of light to find frequency:
u = rac{c}{ ext{λ}} = rac{3 imes 10^8 ext{ m/s}}{5.2 imes 10^{-7} ext{ m}} = 5.77 imes 10^{14} ext{ s}^{-1}.Calculate energy:
E = h
u = (6.626 imes 10^{-34} ext{ J·s})(5.77 imes 10^{14} ext{ s}^{-1}) = 3.82 imes 10^{-19} ext{ J}.
Nature of Light: The Photoelectric Effect
Classical physics could not explain why certain frequencies of light would eject electrons from metal surfaces.
Threshold Frequency ($ u_0$):
Light below this frequency does not result in electron ejection. Light above $
u_0$ increases kinetic energy but not quantity of emitted electrons.
Quantum Mechanics: Einstein’s Nobel Prize
Albert Einstein (1905):
Explained the photoelectric effect by incorporating quantization ideas from Planck, showing light as discrete packets called photons.
Energy of a photon can be expressed as:
E_ ext{photon} = h
u.
Worked Example: Energy of a Photon in a Microwave
Calculate energy of one photon with a wavelength of 1.20 cm:
E = rac{hc}{ ext{λ}} = rac{(6.626 imes 10^{-34} ext{ J·s})(3.00 imes 10^8 ext{ m/s})}{1.20 imes 10^{-2} ext{ m}} = 1.66 imes 10^{-23} ext{ J}.
Atomic Spectra
Emission spectra produced when electric current passes through a gas in a vacuum tube, causing light emission. Each element exhibits its unique spectrum, which can be used for identification.
Applying Quantum Mechanics: The Bohr Atom
Niels Bohr (1912):
Developed a model with quantized energy levels, accurately predicting the hydrogen spectrum.
Electron transitions between orbits lead to spectral lines:
En = - rac{RH}{n^2}, where R_H = 2.18 imes 10^{-18} ext{ J}.
Ground State vs. Excited State
Ground State: Atom has no absorbed energy (most stable configuration).
Excited State: Electrons can reach higher energy states, emitting energy as light upon returning to ground state.
The Quantum-Mechanical Model: Quantum Numbers
Each electron in an atom is described by four quantum numbers:
Principal Quantum Number ($n$): Indicates energy and size of orbit. (Positive integers: 1, 2, …)
Angular Momentum Quantum Number ($ ext{l}$): Determines the shape of the orbital (0, 1, 2, …, $n-1$).
Magnetic Quantum Number ($m_ ext{l}$): Indicates the orientation of the orbital (-$ ext{l}, … , + ext{l}$).
Spin Quantum Number ($m_s$): Represents the spin of the electron (+1/2 or -1/2).
Orbital Nodes
Node: A region with zero probability of finding an electron, arising from wave interference.
Types:
Angular Nodes: Angular distributions (not found in s orbitals).
Radial Nodes: Found in radial distributions; number given by $n - ext{l} - 1$.
The Quantum Field Theory
Paul Dirac (1925): Developed a unified formulation combining wave mechanics and matrix mechanics, which led to the treatment of particles as both waves and particles.
Summary of Quantum Numbers and Orbitals
Four quantum numbers define electron states in multi-electron systems. No two electrons can share the same set of quantum numbers.
Lecture Learning Goals
Recognize distinctions in wave/particle properties of light.
Understand orbital shapes via probability.
Identify nodes and orbital phases.
Distinguish among orbitals and describe configurations in multi-electron atoms or ions.