Chapter 6: Electronic Structure of Atoms – Study Notes

Electronic Structure of Atoms

Introduction to Electronic Structure

The electronic structure of atoms refers to the arrangement and energy of electrons within an atom. Understanding this structure is essential for explaining chemical properties and behaviors. The study of electronic structure begins with the nature of electromagnetic radiation, as electrons exhibit both wave-like and particle-like properties.

The Wave Nature of Light

Electromagnetic Radiation

Electromagnetic radiation consists of oscillating electric and magnetic fields that travel through space at the speed of light. Key characteristics include:

• Wavelength (\(\lambda\)): The distance between corresponding points on adjacent waves.

• Frequency (\(\nu\)): The number of waves passing a given point per unit time.

• For waves traveling at the same velocity, a longer wavelength means a lower frequency, and vice versa.

Relationship Between Wavelength and Frequency

All electromagnetic radiation travels at the speed of light (\(c = 3.00 \times 10^8\) m/s). The relationship between wavelength and frequency is given by:

$c = \lambda \nu$

Where:

• \(c\) = speed of light

• \(\lambda\) = wavelength

• \(\nu\) = frequency

Units of Wavelength

Common units for wavelength include:

• Angstrom (\(\text{\AA}\)): $1 \text{\AA} = 1 \times 10^{-10}$ m (X-rays)

• Nanometer (nm): $1 \text{nm} = 1 \times 10^{-9}$ m (Ultraviolet, visible)

• Micrometer (\(\mu\)m): $1 \mu\text{m} = 1 \times 10^{-6}$ m (Infrared)

• Millimeter (mm): $1 \text{mm} = 1 \times 10^{-3}$ m (Microwave)

• Centimeter (cm): $1 \text{cm} = 1 \times 10^{-2}$ m (Microwave)

• Meter (m): $1$ m (Television, radio)

• Kilometer (km): $1$km = $1000$ m (Radio)

Quantized Energy and Photons

Limitations of the Wave Model

Three phenomena could not be explained by the wave model of light:

1. Blackbody radiation (emission of light from hot objects)

2. Photoelectric effect (emission of electrons from metal surfaces when illuminated)

3. Emission spectra of excited gas atoms

Blackbody Radiation and Quanta

Max Planck proposed that energy is quantized and can only be emitted or absorbed in discrete packets called quanta (singular: quantum). This is analogous to walking up steps (quantized) versus a ramp (continuous).

The Photoelectric Effect

Albert Einstein explained the photoelectric effect by proposing that light consists of particles called photons, each with energy proportional to its frequency:

$E = h\nu$

Where \(h = 6.626 \times 10^{-34}\) J·s is Planck's constant. Electrons are only emitted if the photon energy exceeds a threshold specific to each metal.

Line Spectra and the Bohr Model

Atomic Emission Spectra

When atoms are excited, they emit light at specific wavelengths, producing a line spectrum unique to each element, rather than a continuous spectrum.

The Hydrogen Spectrum and Rydberg Formula

The wavelengths of hydrogen's emission lines can be described by the Rydberg formula:

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

Where \(R_H\) is the Rydberg constant, and \(n_1\) and \(n_2\) are integers with \(n_2 > n_1\).

The Bohr Model of the Atom

Niels Bohr proposed that electrons move in specific, quantized orbits around the nucleus. Key postulates include:

• Only certain orbits with specific energies are allowed.

• Electrons in allowed orbits do not radiate energy.

• Energy is absorbed or emitted only when an electron transitions between orbits, with the energy difference given by:

$\Delta E = E_f - E_i = -2.18 \times 10^{-18} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \text{J}$

Energy States

• Ground state: Lowest energy level (n = 1).

• Excited state: Any energy level with n > 1.

• Positive \(\Delta E\): Energy absorbed (photon absorbed).

• Negative \(\Delta E\): Energy released (photon emitted).

Limitations of the Bohr Model

• Accurately describes only hydrogen (one-electron systems).

• Does not account for electron-electron interactions or the wave nature of electrons.

The Wave Behavior of Matter

de Broglie Hypothesis

Louis de Broglie proposed that particles, such as electrons, have wave properties. The wavelength associated with a particle is given by:

$\lambda = \frac{h}{mv}$

Where \(m\) is mass and \(v\) is velocity.

Heisenberg Uncertainty Principle

Werner Heisenberg showed that it is impossible to know both the position and momentum of a particle with absolute precision:

$\Delta x \cdot m\Delta v \geq \frac{h}{4\pi}$

This principle is fundamental to quantum mechanics.

Quantum Mechanics and Atomic Orbitals

Schrödinger Equation and Wave Functions

Erwin Schrödinger developed a mathematical model (the Schrödinger equation) that describes electrons as wave functions (\(\psi\)). The square of the wave function (\(\psi^2\)) gives the probability density of finding an electron in a particular region.

Quantum Numbers

Each atomic orbital is described by a set of 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 (ml): Orientation in space (ml = -l to +l)

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

Types of Orbitals

• s orbitals (l = 0): Spherical shape

• p orbitals (l = 1): Two lobes with a node between them

• d orbitals (l = 2): Four lobes (except one with a doughnut shape)

• f orbitals (l = 3): Complex shapes

Degeneracy and Energy Levels

For hydrogen, all orbitals with the same principal quantum number (n) have the same energy (degenerate). In multi-electron atoms, energy levels split due to electron-electron repulsion.

Spin Quantum Number and Pauli Exclusion Principle

Each electron has a spin quantum number (ms = +1/2 or -1/2). The Pauli Exclusion Principle states that no two electrons in the same atom can have the same set of four quantum numbers.

Electron Configurations

Writing Electron Configurations

Electron configuration describes the arrangement of electrons in an atom. The notation includes:

• Principal energy level (n)

• Type of orbital (s, p, d, f)

• Number of electrons in the orbital (superscript)

Example: $1s^2 2s^2 2p^6$

Orbital Diagrams and Hund's Rule

Orbital diagrams use boxes and arrows to represent orbitals and electron spins. Hund's Rule states that electrons fill degenerate orbitals singly before pairing up, maximizing the number of electrons with the same spin.

Condensed Electron Configurations

Core electrons are represented by the symbol of the preceding noble gas in brackets. Only valence electrons are written explicitly.

Example: [He]2s1 for Li

Transition Metals, Lanthanides, and Actinides

Transition metals fill d orbitals, while lanthanides and actinides fill f orbitals. Some anomalies occur due to the close energy levels of s, d, and f orbitals.

Electron Configurations and the Periodic Table

Periodic Table Blocks

The periodic table is divided into blocks corresponding to the type of orbital being filled:

• s-block: Groups 1 and 2

• p-block: Groups 13–18

• d-block: Transition metals

• f-block: Lanthanides and actinides

Electron Configuration Anomalies

Some elements (e.g., chromium, copper) have electron configurations that differ from the expected order due to the stability associated with half-filled or fully filled subshells.

Additional info: These notes cover the fundamental concepts of Chapter 6: Electronic Structure of Atoms, including the quantum mechanical model, quantum numbers, and their relationship to the periodic table. All equations are provided in LaTeX format for clarity.