10.2 Quantum numbers and atomic orbitals

Quantum numbers and atomic orbitals are key to understanding electron behavior in atoms. These concepts describe the energy, shape, and orientation of electron orbitals, which determine an atom's properties and bonding capabilities.

By mastering quantum numbers and orbital shapes, you'll grasp how electrons are arranged in atoms. This knowledge is crucial for predicting chemical reactivity, spectral properties, and the periodic trends that shape the elements' behavior.

Quantum Numbers

Principal and Azimuthal Quantum Numbers

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• Principal quantum number (n) determines the energy level and size of an orbital
  • Takes positive integer values (1, 2, 3, ...)
  • Larger n values indicate higher energy levels and larger orbitals
  • Relates to the average distance of an electron from the nucleus
• Azimuthal quantum number (l) describes the shape of the orbital
  • Takes integer values from 0 to n-1
  • Corresponds to subshells (s, p, d, f)
  • s orbital: l = 0, p orbital: l = 1, d orbital: l = 2, f orbital: l = 3
• Relationship between n and l defines allowed combinations of energy levels and orbital shapes
  • 1s orbital: n = 1, l = 0
  • 2p orbital: n = 2, l = 1
  • 3d orbital: n = 3, l = 2

Magnetic and Spin Quantum Numbers

• Magnetic quantum number (m_l) specifies the orbital's orientation in space
  • Takes integer values from -l to +l, including zero
  • Determines the number of orbitals in a subshell (2l + 1)
  • p orbitals (l = 1) have three possible m_l values: -1, 0, +1
• Spin quantum number (m_s) represents the intrinsic angular momentum of an electron
  • Takes values of +1/2 or -1/2
  • Describes the two possible spin states of an electron (often referred to as "up" and "down")
• Pauli exclusion principle states no two electrons in an atom can have the same set of four quantum numbers
  • Limits the number of electrons in each orbital to two with opposite spins

Atomic Orbitals

Orbital Types and Shapes

• s orbitals have a spherical shape centered on the nucleus
  • One orbital per subshell (m_l = 0)
  • Simplest orbital shape with no angular nodes
• p orbitals have a dumbbell shape with two lobes
  • Three orbitals per subshell (m_l = -1, 0, +1)
  • Oriented along x, y, and z axes (px, py, pz)
• d orbitals have more complex shapes with four lobes
  • Five orbitals per subshell (m_l = -2, -1, 0, +1, +2)
  • Four orbitals have a cloverleaf shape, one has a doughnut shape with a cloverleaf
• f orbitals have even more intricate shapes
  • Seven orbitals per subshell (m_l = -3, -2, -1, 0, +1, +2, +3)
  • Complex shapes with multiple lobes and orientations

Probability Distributions and Nodes

• Radial probability distribution describes the likelihood of finding an electron at a certain distance from the nucleus
  • Represented by radial distribution function R(r)
  • Shows peaks and valleys corresponding to regions of high and low electron density
• Angular probability distribution determines the likelihood of finding an electron in a particular direction from the nucleus
  • Represented by spherical harmonics Y(θ,φ)
  • Defines the characteristic shapes of different orbital types
• Node refers to a region where the probability of finding an electron is zero
  • Radial nodes are spherical surfaces where the radial wave function is zero
  • Angular nodes are planes or conical surfaces where the angular wave function is zero
• Number of nodes increases with higher quantum numbers
  • Radial nodes: n - l - 1
  • Angular nodes: l

Orbital Properties

Degeneracy and Energy Levels

• Degeneracy refers to orbitals with the same energy level
  • Occurs when different combinations of quantum numbers result in the same energy
  • Degenerate orbitals have the same n and l values but different m_l values
• Orbital energy primarily depends on the principal quantum number (n)
  • Higher n values correspond to higher energy levels
  • In multi-electron atoms, energy also depends on l due to electron-electron interactions
• Aufbau principle guides electron configuration
  • Electrons fill orbitals from lowest to highest energy
  • Order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p

Orbital Orientation and Spatial Distribution

• Orbital shape determined by the azimuthal quantum number (l)
  • s orbitals (l = 0): spherical
  • p orbitals (l = 1): dumbbell-shaped
  • d orbitals (l = 2): more complex, four-lobed or doughnut shapes
• Orbital orientation in space specified by the magnetic quantum number (m_l)
  • p orbitals: px, py, pz aligned along respective axes
  • d orbitals: various orientations (dxy, dyz, dxz, dx2-y2, dz2)
• Spatial distribution of electrons affects chemical bonding and molecular geometry
  • Directional nature of p and d orbitals influences bond angles and molecular shapes
  • Hybridization of atomic orbitals explains observed molecular structures (sp, sp2, sp3)