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 progresses through quantum mechanical models.

6.1 The Wave Nature of Light

Electromagnetic Radiation

• Electromagnetic radiation consists of energy propagated as waves through space at the speed of light.

• The wavelength ($\lambda$) is the distance between corresponding points on adjacent waves.

• The frequency ($\nu$) is 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.

Equation:

$c = \lambda \nu$

where $c$ is the speed of light ($3.00 \times 10^8$ m/s).

Electromagnetic Spectrum

• Electromagnetic radiation includes a range of wavelengths and frequencies, from gamma rays to radio waves.

• Different regions of the spectrum correspond to different types of radiation (e.g., ultraviolet, visible, infrared).

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

Evidence for Quantization

• Three phenomena could not be explained by classical wave theory:

  1. Blackbody radiation: Emission of light from hot objects.

  2. Photoelectric effect: Emission of electrons from metal surfaces when illuminated by light.

  3. Emission spectra: Light emitted from electronically excited gas atoms.

Blackbody Radiation

• Objects emit light when heated, but classical physics could not explain the observed spectrum.

• Max Planck proposed that energy is emitted or absorbed in discrete packets called quanta (singular: quantum).

• This quantization is analogous to climbing stairs (discrete steps) rather than walking up a ramp (continuous).

The Photoelectric Effect

• Einstein explained the photoelectric effect using the concept of quanta (photons).

• Electrons are ejected from a metal only if the incident light has a frequency above a certain threshold.

• The energy of a photon is proportional to its frequency:

$E = h\nu$

where $h$ is Planck's constant ($6.626 \times 10^{-34}$ J·s).

Atomic Emission Spectra

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

6.3 Line Spectra and the Bohr Model

Line Spectra

• Unlike a continuous spectrum, a line spectrum consists of discrete wavelengths.

• Each element has a unique line spectrum, which can be used for identification.

The Hydrogen Spectrum

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

• The Rydberg formula 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

• Niels Bohr proposed that electrons move in specific orbits with quantized energies.

• Key postulates:

  1. Only certain orbits with specific energies are allowed for electrons in a hydrogen atom.

  2. An electron in an allowed orbit does not radiate energy.

  3. Energy is emitted or absorbed only when an electron changes orbits, with the energy given by $E = h\nu$.

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

• The energy change for a transition is:

$\Delta E = E_{n_2} - E_{n_1} = -2.18 \times 10^{-18} \left( \frac{1}{n_2^2} - \frac{1}{n_1^2} \right) \text{J}$

• A positive $\Delta E$ means energy is absorbed (photon absorbed, $n_2 > n_1$).

• A negative $\Delta E$ means energy is released (photon emitted, $n_2 < n_1$).

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.

Key Ideas Incorporated into Modern Theory

• Electrons exist in discrete energy levels described by quantum numbers.

• Energy is involved in transitions between these levels.

6.4 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.

The Uncertainty Principle

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

• The mathematical expression is:

$(\Delta x)(\Delta mv) \geq \frac{h}{4\pi}$

6.5 Quantum Mechanics and Atomic Orbitals

Schrödinger Equation and Quantum Mechanics

• Erwin Schrödinger developed a mathematical model (quantum mechanics) that incorporates both wave and particle nature of matter.

• The solutions to the Schrödinger equation are called wave functions ($\psi$).

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

Atomic Orbitals and Quantum Numbers

• Each orbital is described by a set of three quantum numbers: $n$, $l$, $m_l$.

• Principal quantum number ($n$): Indicates energy level and size of the orbital (values: 1, 2, 3, ...).

• Angular momentum quantum number ($l$): Defines the shape of the orbital (values: 0 to $n-1$).

• Magnetic quantum number ($m_l$): Specifies the orientation of the orbital (values: $-l$ to $+l$).

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

Summary Table: Quantum Numbers and Orbitals (up to $n=4$)

n	l	Subshell	ml	Number of Orbitals
1	0	1s	0	1
2	0	2s	0	1
2	1	2p	-1, 0, 1	3
3	0	3s	0	1
3	1	3p	-1, 0, 1	3
3	2	3d	-2, -1, 0, 1, 2	5
4	0	4s	0	1
4	1	4p	-1, 0, 1	3
4	2	4d	-2, -1, 0, 1, 2	5
4	3	4f	-3, -2, -1, 0, 1, 2, 3	7

6.6 Representation of Orbitals

s Orbitals

• For $l=0$, orbitals are spherical in shape.

• The radius increases with increasing $n$.

• For an $ns$ orbital, the number of peaks is $n$; the number of nodes is $n-1$.

p Orbitals

• For $l=1$, p orbitals have two lobes with a node at the nucleus.

d Orbitals

• For $l=2$, four of the five d orbitals have four lobes; one has a doughnut-shaped ring around the center.

f Orbitals

• For $l=3$, f orbitals have very complex shapes (not shown in basic texts).

• There are seven equivalent f orbitals in a sublevel.

6.7 Hydrogen Atom Orbital Energies and Many-Electron Atoms

Hydrogen Atom

• For a one-electron hydrogen atom, all orbitals with the same $n$ have the same energy (they are degenerate).

Many-Electron Atoms

• Electron-electron repulsion causes splitting of energy levels; not all orbitals with the same $n$ are degenerate.

• Within a subshell, orbitals remain degenerate.

• Energy levels can overlap (e.g., 4s is lower in energy than 3d).

Additional info: These concepts form the foundation for understanding electron configurations, periodic trends, and chemical bonding in subsequent chapters.