Quantum-Mechanical Model of the Atom
Expectations in Understanding the Quantum-Mechanical Model
Work with wavelength, frequency, and energy of electromagnetic radiation.
Know the order of the regions in the electromagnetic spectrum based on energy and wavelength.
Interpret line spectra of elements through laboratory exercises.
Understand electronic structure including:
Quantum numbers
Orbitals
Electron configurations (next unit, but linked to current topics).
The Periodic Table and Electronic Structure
The periodic table is crucial for understanding electronic structure.
Group classifications:
1A: Alkali metals
2A: Alkaline earth metals
3A-8A: p-block
Transition metals and others, characterized by various electron configurations.
The Wave Nature of Light
Definition: Electromagnetic radiation is a form of energy including light, heat, microwaves, and radio waves.
Speed of Light: In a vacuum, the speed of light is given as:
c = 2.9979 imes 10^8 ext{ m/s}Wave Characteristics:
Wavelength (λ) measured in meters (m) or nanometers (nm)
Frequency (ν) measured in Hertz (Hz) or s$^{-1}$
Velocity (c) in m/s
Amplitude: Height of a wave
Node: A point where amplitude equals zero.
Color and Brightness
Different wavelengths result in different colors of light.
Different amplitudes influence brightness, charting a spectrum from dim to bright light.
Description of Electromagnetic Radiation
Electromagnetic radiation is described as a wave composed of oscillating electric and magnetic fields in perpendicular planes.
Frequency: Defined as the number of waves that pass a specified point in a specified time frame, with low frequencies corresponding to longer wavelengths and high frequencies to shorter wavelengths.
Electromagnetic Spectrum Overview
The electromagnetic spectrum categorizes radiation by frequency (ν) and wavelength (λ). Main types include:
Radio waves
Microwaves
Infrared radiation
Visible light
Ultraviolet radiation
X-rays
Gamma rays
Units for the various types of radiation include:
Type of Radiation
Length (m)
Angstrom (Å)
$10^{-10}$
Nanometer (nm)
$10^{-9}$
Micrometer (μm)
$10^{-6}$
Millimeter (mm)
$10^{-3}$
Centimeter (cm)
$10^{-2}$
Meter (m)
$1$
Kilometer (km)
$10^{3}$
Relationship between Wavelength and Frequency
The speed of light is a constant:
c =
u imes ext{λ}Thus,
As wavelength (λ) increases, frequency (ν) decreases, and vice versa.
Visible Light Wavelength Range: 350 – 760 nm; arrangement should be known.
Interference and Diffraction of Waves
Interference: Interaction between waves resulting in:
Constructive Interference: Waves add to form a larger amplitude (in phase).
Destructive Interference: Waves cancel each other (out of phase).
Diffraction: Occurs when waves encounter openings or obstacles comparable in size to their wavelength, resulting in bending. This is characteristic of wave propagation and does not apply to traveling particles.
Two-Slit Interference Experiment
Description of light behavior when it passes through two small slits, yielding:
An interference pattern where constructive interference (equal path lengths) produces bright spots and destructive interference (path lengths differing by half) produces dark spots.
Quantized Energy and Photons
Historical Context: Initially, matter and energy were treated as unrelated until significant discoveries in 1900.
Key problems addressed:
Blackbody radiation emission
Photoelectric effect
Emission from excited gas atoms (emission spectra)
Max Planck’s Contribution: Proposed that energy of matter is quantized, existing in discrete units defined by:
E_{quantum} = h
u
Where:$h = 6.626 imes 10^{-34} ext{ J·s}$ (Planck’s constant)
Albertn Einstein’s Contribution: Explained electromagnetic radiation as quantized packets called photons, where the energy of a photon is determined by:
E_{photon} = h
u
The Photoelectric Effect
Key observations include: A threshold frequency is required for electron emission, regardless of intensity.
Einstein posited that light energy comes in packets (quanta or photons), with photon energy directly proportional to frequency and inversely proportional to wavelength.
Energy Calculations
Example Problem: Calculate the energy needed for photonic emission with a known threshold frequency.
Energy for a Photon: Must be quantified by the formula E = h
u and extended to a mole of photons using Avogadro's number.
Emission Spectra and Bohr Model
Spectrum Types:
Continuous Spectrum: Contains all wavelengths.
Line Spectrum: Contains only specific wavelengths relevant to specific elements.
Gases produce unique line spectra when evaluated under specific conditions.
Rydberg Equation**
Utilized to calculate the wavelengths for hydrogen’s line spectrum:
ext{Rydberg Equation}: rac{1}{ ext{λ}} = RHigg( rac{1}{nf^2} - rac{1}{ni^2}igg) Where $ni$ and $n_f$ are integers representing principal quantum numbers.
Bohr Model of the Atom
Postulates:
Electrons occupy specific energy levels corresponding to designated orbits.
Energy transitions occur when electrons move between these levels, resulting in emission or absorption of energy.
Energy in Hydrogen Atom:
The energy of each orbit is quantified, with negative values indicating stable configurations.
The ground state is at $n=1$ and excited states are defined for higher integer values up to infinity.
Electronic Transitions and Spectroscopy
Electronic transitions are defined based on the type of light emitted:
Ultraviolet (Lyman series): $ni o nf = 1$
Visible (Balmer series): $ni o nf = 2$
Infrared (Paschen series): $ni o nf = 3$
Energy Difference Calculation: Use the equation ext{ΔE} = -2.18 imes 10^{-18} ext{ J} igg( rac{1}{nf^2} - rac{1}{ni^2}igg) for transitions between states.
Limitations of the Bohr Model
While effective for hydrogen, the model failed for multi-electron systems.
Reinforced the understanding that electrons are constrained to energy levels without orbiting the nucleus in fixed paths.
Quantum Mechanics and Matter
Louis de Broglie: Suggested that particles like electrons exhibit wave-like properties, leading to calculations of wavelength given mass:
ext{λ} = rac{h}{mv}Heisenberg Uncertainty Principle: Established that both position and momentum of a particle cannot be known simultaneously.
Quantum Mechanics and Atomic Orbitals
The Schrödinger equation treated electrons as waves and identified regions of high likelihood for finding electrons, called orbitals.
Each orbital’s characterization utilizes four quantum numbers:
Principal Quantum Number ($n$): Defines the size and energy.
Angular Momentum Quantum Number ($l$): Defines shape (s, p, d, f orbitals).
Magnetic Quantum Number ($m_l$): Defines orientation of the orbital.
Spin Quantum Number ($m_s$): Defines electron spin states (up or down).
Probability Densities and Orbitals
Different orbitals (s, p, d, f) exemplified through density functions, with nodes indicating regions of zero probability.
Radial distribution functions display probabilities while accounting for distances from the nucleus, showcasing behaviors of electrons in different energy states.
Electron Behavior and Chemical Bonding
Interactions of phase between orbitals dictate bonding characteristics.
Atomic structures are illustrated through collective orbitals forming spherical shapes due to their probability densities.