Gen Chem Exam 3

Chapter 6: Electronic Structure and Periodic Properties of Elements

Homework

Due: March 28, 2026, by 11:59 pm
Assignment: Homework 5

Crab Nebula and Spectral Analysis

The Crab Nebula consists of remnants of a supernova (the explosion of a star).
NASA’s Hubble Space Telescope produced a composite image of the Nebula.
Measurements of the emitted light wavelengths enabled astronomers to identify elements in the nebula.
Specific ions identified include:
- $S^+$ (green filaments)
- $O^{2+}$ (red filaments)

Wave Properties of Light

Light can exhibit wave properties as well as particle properties.
Key terms and sections:
- Magnetic Field
- Direction: Represented along the y-axis in diagrams.
- Electric Field
- Direction: Represented along the z-axis in diagrams.
- Propagation Direction
- Represented along the x-axis in diagrams.

Characteristics of Waves

All waves are characterized by the following properties:
1. Wavelength (λ)
- Units: meters (m)
- Definition: Distance between two consecutive peaks or troughs in a wave.
1. Frequency (ν)
- Units: Hertz (Hz) or $s^{-1}$
- Definition: Number of successive wavelengths that pass a given point in a unit of time.
1. Amplitude
- Units: Volts per meter (V/m)
- Definition: One-half the distance between the peaks and troughs of the wave.

Relationship Between Wavelength, Frequency, and Speed of Light

The product of a wave's wavelength (λ) and its frequency (ν) is the speed of the wave.
For electromagnetic radiation in a vacuum:
- Relationship: $c = ext{λν}$
- Where: $c = 2.998 imes 10^8 ext{ m/s}$

Electromagnetic Spectrum

The electromagnetic spectrum can be organized based on increasing energy (E), frequency, and wavelength.
Parts of the spectrum include:
- Radio Waves
- Microwaves
- Infrared Radiation
- Visible Light
- Ultraviolet Radiation
- X-rays
- Gamma Rays

Example Calculation of Light Frequency

Given:
- Wavelength of sodium street light: 589 nm.
- Conversion Required: $1 ext{ nm} = 1 imes 10^{-9} ext{ m}$ .
Calculation:
- $c = ext{λν}$ . Rearranging gives:
- $ν = rac{c}{λ}$
- Example: $ν = rac{2.998 imes 10^8 ext{ m/s}}{589 imes 10^{-9} ext{ m}} = 5.09 imes 10^{14} s^{-1} ext{ (Hz)}$ .

Standing Waves

Definition: Standing waves (or stationary waves) remain constrained within some region of space.
Importance: Playing a role in the understanding of the electronic structure of atoms and molecules.
Example: A vibrating string fixed at both ends.

Interference of Light

Constructive Interference: Occurs when crests from two waves align, increasing amplitude.
Destructive Interference: Occurs when a crest and trough align, canceling each other out.
Demonstrates that light behaves as a wave.

Blackbody Radiation

Describes how temperatures relate to a range of wavelengths.
Predicted infinite energy at high frequencies, leading to the ultraviolet catastrophe (failed predictions about UV light).

Max Planck and Quantized Energy

Planck's formula for quantized energy:
- $E = nhν$
- Where:
- $E$ = energy
- $n$ = integer
- $h$ = Planck’s constant, h = 6.626 imes 10^{-34} ext{ J} ullet ext{s}
Planck's achievements around 1900 included:
- Theoretical expressions fitting experimental data precisely.
- Concept that atoms vibrate at varying frequencies (or wavelengths) as temperature increases.

Photoelectric Effect

Electrons can be ejected from metal surfaces when illuminated by light with sufficient frequency (greater than the threshold frequency).
Observations:
- Kinetic energy of ejected electrons correlates with light frequency, not intensity.
Light viewed as particles (photons): Albert Einstein applied Planck's findings to address the wave-particle duality.

Line Spectra and Rydberg Equation

Emission lines consist of discrete wavelengths of light.
Rydberg Equation:
- Predicts hydrogen's emission lines:
- Formula: rac{1}{λ} = R_∞igg( rac{1}{n_1^2} - rac{1}{n_2^2}igg), where $n_1$ and $n_2$ are integers (with n_1 < n_2) and $R_∞$ is the Rydberg constant, $R_∞ = 1.097 imes 10^7 ext{ m}^{-1}$ .

Bohr Model and Quantum Mechanics

Integrates:
- Planck's quantization principles.
- Insights from Einstein on the particle nature of light.
Bohr's Assumptions: Atoms consist of dense nuclei with orbiting electrons.
- Transitions in electron location lead to absorption/emission of photons.

Summary of Quantum Concepts

Bohr: Electrons occupy quantized energy levels.
Einstein: Light behaves as particles (photons).
De Broglie: Matter such as electrons can exhibit wave properties.
Heisenberg: Uncertainty in simultaneous measurement of momentum and position of a particle.
Schrödinger: Electrons defined by wavefunctions in 3D space.
Born: Wavefunction describes the probability of finding electrons.

Quantum Numbers

Three key quantum numbers:
- Principle (n): Defines size and energy level.
- Secondary (ℓ): Defines orbital shape; values range from 0 to $n-1$ .
- Magnetic (mℓ): Defines orientation; values range from $- ext{ℓ}$ to $+ ext{ℓ}$ .
- Spin (ms): Defines electron spin direction; values of $+ rac{1}{2}$ or $- rac{1}{2}$ .

Electron Configurations and the Periodic Table

Valence Electrons: Electrons in the outermost shell.
Core Electrons: Inner shell electrons.
Configurations can be abbreviated using noble gas references.

Trends in Atomic Radius, Ionization Energy, and Electron Affinity

Atomic Radius Trends: Increases down a group, decreases across a period.
Ionization Energy Trends: Decrease down a group, increase across a period.
Electron Affinity: Energy change when an electron is added to a gaseous atom, can be exothermic (negative) or endothermic (positive).