Quantum-Mechanical Model of the Atom

The Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, which differ in wavelength and frequency. The energy of radiation increases as the frequency increases and the wavelength decreases.

• Regions: Includes (from low to high energy): radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.

• Relative Energies: Gamma rays have the highest energy, while radio waves have the lowest.

• Visible Light: The small portion of the spectrum visible to the human eye (approximately 400–700 nm).

Wavelength and Frequency Conversion

Wavelength (λ) and frequency (ν) are inversely related for electromagnetic waves, connected by the speed of light (c).

• Equation:

• Speed of light (c): m/s

• Example: If a wave has a frequency of Hz, its wavelength is m (500 nm).

Energy and Frequency (Planck's Equation)

The energy of a photon is directly proportional to its frequency, as described by Planck's equation.

• Equation:

• Planck's constant (h): J·s

• Example: A photon with frequency Hz has energy J.

Particle-Wave Duality and de Broglie Equation

All matter exhibits both particle and wave properties. The de Broglie equation relates a particle's momentum to its wavelength.

• Equation:

• m: mass of the particle (kg)

• v: velocity of the particle (m/s)

• Example: An electron (mass kg) moving at m/s has m.

Electron Transitions: Balmer-Rydberg Equation

The wavelength and energy of light emitted or absorbed during electron transitions in a hydrogen atom can be calculated using the Balmer-Rydberg equations.

• Wavelength Equation:

• Energy Equation:

• Rydberg constant (for energy): J

• Rydberg constant (for wavelength): m

• ni: initial energy level; nf: final energy level

• Example: Calculate the wavelength of light emitted when an electron falls from to in hydrogen.

Heisenberg Uncertainty Principle

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

• Equation:

• Implication: The more precisely one property is known, the less precisely the other can be known.

Atomic Orbitals and Nodes

Orbitals are regions in an atom where there is a high probability of finding an electron. Nodes are regions where the probability of finding an electron is zero.

• s orbitals: Spherical in shape; have n-1 nodes.

• p orbitals: Dumbbell-shaped; have n-2 nodes.

• Example: The 2p orbital has one node.

Electron Configuration and Quantum Numbers

Aufbau Principle and Related Rules

Electrons fill atomic orbitals in a specific order, governed by several principles:

• Aufbau Principle: Electrons occupy the lowest energy orbitals first.

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

• Hund’s Rule: Electrons fill degenerate (equal energy) orbitals singly before pairing up.

Assigning Electron Configurations

Electron configurations describe the arrangement of electrons in an atom or ion.

• Full configuration: Lists all occupied orbitals (e.g., 1s2 2s2 2p6).

• Noble gas shorthand: Uses the previous noble gas to abbreviate (e.g., [Ne] 3s2 3p5).

• Orbital diagrams: Visual representations using arrows to show electron spins in orbitals.

• Quantum numbers: Each electron is described by four quantum numbers: principal (n), angular momentum (l), magnetic (ml), and spin (ms).

Valence Shell Electrons

Valence electrons are those in the highest principal energy level of an atom. They are primarily responsible for chemical bonding and reactivity.

Periodic Properties of the Elements

Effective Nuclear Charge (Zeff)

The effective nuclear charge is the net positive charge experienced by valence electrons. It generally increases from left to right across a period due to increasing nuclear charge with minimal additional shielding.

Atomic Radius

The atomic radius is the distance from the nucleus to the outermost electron shell.

• Trend across a period: Decreases from left to right due to increasing Zeff.

• Trend down a group: Increases from top to bottom as additional electron shells are added.

Ionization Energy (Ei)

Ionization energy is the energy required to remove an electron from a gaseous atom.

• Trend across a period: Increases from left to right.

• Trend up a group: Increases from bottom to top.

• Valence vs. core electrons: Ionization energies are much lower for valence electrons than for core electrons.

Electron Affinity (Eea)

Electron affinity is the energy change when an atom gains an electron.

• Trend across a period: Generally becomes more negative (increases) from left to right.

• Trend up a group: Generally becomes more negative from bottom to top.

Octet Rule and Ion Formation

Atoms tend to gain, lose, or share electrons to achieve a full valence shell, often becoming isoelectronic with noble gases. This is known as the octet rule.

• Isoelectronic: Ions with the same electron configuration as a noble gas.

• Example: Na+ and Cl- both have the same electron configuration as Ne and Ar, respectively.