Chapter 6: Electronic Structure of Atoms – Study Notes

Electronic Structure of Atoms

Introduction

The electronic structure of atoms refers to the arrangement and energy of electrons within an atom. Understanding this structure is fundamental to explaining chemical properties and behaviors. The study of electronic structure begins with the nature of electromagnetic radiation and its interaction with matter.

6.1 The Wave Nature of Light

Electromagnetic Radiation

• Electromagnetic radiation is a form of energy that moves as waves through space at the speed of light (c = 3.00 × 108 m/s).

• The wavelength (λ) is the distance between corresponding points on adjacent waves.

• The frequency (ν) is the number of waves passing a given point per unit time.

• Wavelength and frequency are related by the equation: $c = \lambda \nu$

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

Types of Electromagnetic Radiation

• Electromagnetic radiation includes gamma rays, X-rays, ultraviolet, visible light, infrared, microwaves, and radio waves.

• Each type has characteristic wavelengths and energies.

Unit	Symbol	Length (m)	Type of Radiation
Angstrom	Å	10-10	X ray
Nanometer	nm	10-9	Ultraviolet, visible
Micrometer	μm	10-6	Infrared
Millimeter	mm	10-3	Microwave
Centimeter	cm	10-2	Microwave
Meter	m	1	Television, radio
Kilometer	km	1000	Radio

6.2 Quantized Energy and Photons

Blackbody Radiation and Quanta

• Objects emit light when heated (blackbody radiation), which classical physics could not explain.

• Max Planck proposed that energy is quantized and comes in discrete packets called quanta (singular: quantum).

• The energy of a quantum is given by: $E = h\nu$ where $h = 6.626 \times 10^{-34}$ J·s (Planck's constant)

The Photoelectric Effect

• Einstein explained the photoelectric effect by proposing that light consists of particles called photons.

• Each photon has energy $E = h\nu$.

• Electrons are ejected from metal surfaces only if the photon energy exceeds a threshold value specific to the metal.

Atomic Emission and Line Spectra

• Atoms and molecules emit light at specific wavelengths, producing a line spectrum unique to each element.

• This phenomenon could not be explained by classical wave theory.

6.3 Line Spectra and the Bohr Model

The Hydrogen Spectrum

• Johann Balmer and Johannes Rydberg developed formulas relating the observed emission lines of hydrogen to integers.

• The Rydberg formula for the wavelengths of hydrogen emission lines is: $\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$ where $R_H$ is the Rydberg constant.

The Bohr Model of the Atom

• Electrons move in orbits of specific radii and energies around the nucleus.

• Only certain orbits (energy levels) are allowed.

• Energy is absorbed or emitted when an electron transitions between energy levels: $\Delta E = E_f - E_i = -2.18 \times 10^{-18} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$ J

• The lowest energy state ($n=1$) is the ground state; higher energy states ($n>1$) are excited states.

• A positive $\Delta E$ means energy is absorbed (photon absorbed); a negative $\Delta E$ means energy is released (photon emitted).

Limitations of the Bohr Model

• Works only for hydrogen (one-electron systems).

• Does not account for electron wave properties or explain why electrons do not spiral into the nucleus.

6.4 The Wave Behavior of Matter

de Broglie Hypothesis

• Louis de Broglie proposed that matter, like light, exhibits wave properties.

• The wavelength of a particle is given by: $\lambda = \frac{h}{mv}$ where $m$ is mass and $v$ is velocity.

The Uncertainty Principle

• Heisenberg's Uncertainty Principle states that it is impossible to know both the exact position and momentum of a particle simultaneously.

• The relationship is given by: $(\Delta x)(\Delta mv) \geq \frac{h}{4\pi}$

6.5 Quantum Mechanics and Atomic Orbitals

Schrödinger Equation and Wave Functions

• Erwin Schrödinger developed quantum mechanics, a mathematical framework incorporating both wave and particle nature of matter.

• The solution to Schrödinger's equation yields wave functions ($\psi$) for electrons.

• The square of the wave function ($\psi^2$) gives the electron density, or the probability of finding an electron at a given location.

Quantum Numbers

• Each orbital is described by three quantum numbers: principal ($n$), angular momentum ($l$), and magnetic ($m_l$).

• These quantum numbers define the size, shape, and orientation of orbitals.

Principal Quantum Number ($n$)

• Describes the energy level and size of the orbital.

• Possible values: 1, 2, 3, ...

• As $n$ increases, orbitals become larger and electrons are less tightly bound.

Angular Momentum Quantum Number ($l$)

• Defines the shape of the orbital.

• Possible values: integers from 0 to $n-1$.

• Letter designations: s ($l=0$), p ($l=1$), d ($l=2$), f ($l=3$).

Value of l	0	1	2	3
Letter used	s	p	d	f

Magnetic Quantum Number ($m_l$)

• Describes the orientation of the orbital in space.

• Possible values: integers from $-l$ to $+l$, including 0.

• For example, for $l=1$ (p orbital), $m_l = -1, 0, +1$.

Summary Table: Quantum Numbers and Orbitals

n	Subshell Designation	Possible l	Possible ml	Number of Orbitals in Subshell	Total Number of Orbitals in Shell
1	1s	0	0	1	1
2	2s, 2p	0, 1	0; -1, 0, 1	1; 3	4
3	3s, 3p, 3d	0, 1, 2	0; -1, 0, 1; -2, -1, 0, 1, 2	1; 3; 5	9
4	4s, 4p, 4d, 4f	0, 1, 2, 3	0; -1, 0, 1; -2, -1, 0, 1, 2; -3, -2, -1, 0, 1, 2, 3	1; 3; 5; 7	16

6.6 Representation of Orbitals

s Orbitals

• For s orbitals ($l=0$), the shape is spherical.

• The number of peaks in the radial probability distribution is $n$.

• The number of nodes (regions of zero probability) is $n-1$.

• The radius of the sphere increases with $n$.

p Orbitals

• For p orbitals ($l=1$), there are three orientations ($m_l = -1, 0, +1$).

• Shape: two lobes with a node at the nucleus.

d and f Orbitals

• d orbitals ($l=2$): five orientations, typically four-lobed shapes.

• f orbitals ($l=3$): seven orientations, more complex shapes.

Additional info: These notes cover the foundational quantum mechanical model of the atom, which is essential for understanding chemical bonding, periodic trends, and spectroscopy in later chapters.