Atomic Energy Levels and Electronic Structure: Study Notes for General Chemistry

Atomic Energy Levels and Electronic Structure

Introduction

This chapter explores the fundamental concepts of atomic energy levels, the properties of light and electrons, quantum numbers, atomic orbitals, and the structure of the periodic table. Understanding these topics is essential for grasping how atoms interact, form chemical bonds, and exhibit unique chemical properties.

Characteristics of Light

Electromagnetic Radiation

• Electromagnetic radiation is energy transmitted through space as waves, including visible light, radio waves, X-rays, and more.

• Light exhibits both wave-like and particle-like properties.

Wave Properties

• Wavelength (λ): The distance between two consecutive peaks of a wave.

• Frequency (ν): The number of wave cycles passing a point per second.

• Amplitude: The height of the wave, related to the intensity of light.

• Speed of light (c): $c = 3.00 \times 10^8 \text{ m/s}$

• Relationship: $c = \lambda \nu$

Particle Properties

• Light can behave as particles called photons.

• Photon energy: $E = h\nu$ where $h = 6.626 \times 10^{-34} \text{ J·s}$ (Planck's constant).

• Photoelectric effect: Electrons are ejected from a metal surface when exposed to light above a threshold frequency.

Electromagnetic Spectrum

• Ranges from radio waves (long wavelength, low frequency) to gamma rays (short wavelength, high frequency).

• Visible light is a small portion of the spectrum (about 400–700 nm).

Properties of Electrons

Wave-Particle Duality

• Electrons exhibit both wave-like and particle-like behavior.

• de Broglie wavelength: $\lambda = \frac{h}{mv}$, where $m$ is mass and $v$ is velocity.

• Electron diffraction experiments confirm wave properties.

Quantisation and Quantum Numbers

Principal Quantum Number (n)

• Defines the main energy level or shell of an electron in an atom.

• Possible values: $n = 1, 2, 3, ...$

Azimuthal Quantum Number (l)

• Defines the shape of the orbital (subshell).

• Possible values: $l = 0, 1, ..., n-1$

• Orbital types: $l = 0$ (s), $l = 1$ (p), $l = 2$ (d), $l = 3$ (f)

Magnetic Quantum Number (m_l)

• Defines the orientation of the orbital in space.

• Possible values: $m_l = -l, ..., 0, ..., +l$

Spin Quantum Number (m_s)

• Describes the spin of the electron.

• Possible values: $m_s = +\frac{1}{2}, -\frac{1}{2}$

Summary Table: Quantum Numbers

Quantum Number	Symbol	Allowed Values	Physical Meaning
Principal	n	1, 2, 3, ...	Energy level (shell)
Azimuthal	l	0 to n-1	Orbital shape (subshell)
Magnetic	m_l	-l to +l	Orbital orientation
Spin	m_s	+1/2, -1/2	Electron spin direction

Atomic Orbitals and Electron Distributions

Orbital Electron Distributions

• Orbitals are regions in space where electrons are likely to be found.

• Shapes: s (spherical), p (dumbbell), d (cloverleaf), f (complex).

• Electron density plots and boundary surfaces illustrate orbital shapes.

Orbital Energies

• Energy levels are quantized; electrons occupy discrete energy states.

• For hydrogen: $E_n = -2.18 \times 10^{-18} \left( \frac{1}{n^2} \right) \text{ J}$

• Energy difference for transitions: $\Delta E = E_{n_f} - E_{n_i}$

Shielding and Effective Nuclear Charge

• Electrons in inner shells shield outer electrons from the full nuclear charge.

• Effective nuclear charge ($Z_{eff}$) influences orbital energies and atomic size.

Structure of the Periodic Table

Periodic Table Organization

• Elements are arranged by increasing atomic number.

• Groups (columns) share similar chemical properties.

• Periods (rows) correspond to principal energy levels.

Aufbau Principle and Orbital Filling Order

• Electrons fill orbitals in order of increasing energy.

• Order: 1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → 5s → 4d → 5p → 6s → 4f → 5d → 6p → 7s → 5f → 6d → 7p

• Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers.

Absorption and Emission Spectra

Atomic Spectra

• Atoms absorb and emit light at specific wavelengths, producing line spectra.

• Each element has a unique emission and absorption spectrum.

• Transitions between energy levels correspond to photon absorption or emission.

Key Equations

• Speed of light: $c = \lambda \nu$

• Photon energy: $E = h\nu$

• de Broglie wavelength: $\lambda = \frac{h}{mv}$

• Hydrogen atom energy levels: $E_n = -2.18 \times 10^{-18} \left( \frac{1}{n^2} \right) \text{ J}$

• Energy change for transitions: $\Delta E = E_{n_f} - E_{n_i}$

Example: Calculating Photon Energy

• Given wavelength $\lambda = 650 \text{ nm}$, calculate energy:

• Convert to meters: $\lambda = 6.50 \times 10^{-7} \text{ m}$

• Frequency: $\nu = \frac{c}{\lambda}$

• Energy: $E = h\nu$

Summary Table: Restrictions on Quantum Numbers

Quantum Number	Symbol	Range
Principal	n	1, 2, 3, ...
Azimuthal	l	0 to n-1
Magnetic	m_l	-l to +l
Spin	m_s	+1/2, -1/2

Conclusion

Understanding atomic energy levels, quantum numbers, and electron distributions is essential for explaining the chemical behavior of elements and the structure of the periodic table. These concepts form the foundation for further study in chemical bonding, molecular structure, and periodic properties.