CHEM 201: Structure and Bonding Lecture Notes

Introduction to Chemistry 201: Structure and Bonding

• The course focuses on representing the structure of chemical species to reflect their properties.

• Key learning areas include:

  • Electronic structure: Core and valence electrons, quantum numbers, light & energy, Molecular Orbital (MO) Theory, electronic structure of atoms (e.g., \text{CO}_2 ).

  • Periodic properties of the elements.

  • Bonding between atoms in a single chemical species: Ionic, covalent, Lewis diagrams, bond energy, bond dipoles, Valence Shell Electron Pair Repulsion (VSEPR), Valence Bond (VB) Theory, resonance.

  • Intermolecular forces.

  • Chemical reactivity.

  • Physical properties of matter.

• Distinguishing Chemical Entities:

  • Atoms: Individual elements (e.g., C, H, O).

  • Chemical Species: Molecules or ions (e.g., \text{H}_2\text{C=O} - formaldehyde, which is 37%). Molecular dipoles are relevant here.

  • Collections of Chemical Species.

• Isomers: Different representations for chemical species (2D vs. 3D).

• From Orbits to Orbitals: Chapter 1 of the online textbook.

Early Atomic Models: Bohr Model

• Niels Bohr (Nobel Prize 1922): Developed a theoretical model for the hydrogen atom.

• Model Description: An electron moves around the nucleus in a fixed path or "orbit," analogous to planets orbiting the sun.

• Energy Levels & Shells:

  • Electrons occupy specific, fixed paths or "orbits."

  • Each orbit has a distinct energy level, labeled by a principal quantum number ( n = 1, 2, 3, \ldots ).

  • Lower n values mean lower energy, closer to the nucleus, and greater stability.

  • Higher n values mean higher energy, further from the nucleus, and less stability.

• An electron in orbit n = \infty is considered "unbound"; it has escaped from the nucleus.

• Failure of Bohr Model:

  • Could only explain the hydrogen atom; failed for atoms with more than one electron.

  • Contributions: Despite its limitations, it was a crucial step forward by introducing:

  1. The concept of the quantum number n .

  2. The idea of fixed energy levels for electrons.

Wave-Particle Duality and Quantum Mechanics

De Broglie's Hypothesis

• Louis de Broglie (Nobel Prize 1929): Proposed that particles, especially electrons within an atom, exhibit both particle and wavelike characteristics.

• Electron Behavior: Electrons with mass m and speed v bound inside an atom behave like standing waves.

• De Broglie Wavelength: Describes the wavelength ( \lambda ) of a particle: \lambda = \frac{h}{mv} . This illustrates how particle behavior (mass m , speed v ) is linked to wave behavior (wavelength \lambda ).

Wave Properties and Terminology (Self-Learning)

• Wave Cycle: One complete oscillation.

• Frequency ( \nu ): The number of cycles per second.

  • Unit: \text{s}^{-1} or Hz (Hertz).

• Wavelength ( \lambda ): The distance between two successive peaks or from the beginning to the end of a cycle.

  • Unit: m, cm, nm, or Å (angstrom).

  • Conversions: 1 \ \text{cm} = 10^{-2} \ \text{m} , 1 \ \text{nm} = 10^{-9} \ \text{m} , 1 \ \text{Å} = 10^{-10} \ \text{m} .

• Amplitude: The maximum height of a wave.

  • Higher amplitude corresponds to more intense radiation.

• Node: A point with zero amplitude where the phase ( \pm signs) of the wave changes.

• Constructive Interference: When two in-phase waves combine, their amplitudes add up, resulting in a wave with larger amplitude.

• Destructive Interference: When two out-of-phase waves combine, their amplitudes cancel each other out, resulting in a wave with smaller or zero amplitude.

Heisenberg Uncertainty Principle

• Werner Heisenberg (Nobel Prize 1932): States that it is impossible to precisely know both the exact position and the exact momentum (mass x speed) of a particle, such as an electron, at the same time.

• Mathematical Expression: \Delta x \cdot \Delta v > \frac{h}{4 \pi m} . This means we cannot precisely know both an electron's exact position ( \Delta x ) and momentum/speed ( \Delta v ) at the same time.

• Implication: Instead of exact position and energy, we calculate the probability of finding an electron of a given energy within a specific space.

Schrödinger Equation and Wave Functions (Orbitals)

• Erwin Schrödinger (Nobel Prize 1933): Introduced mathematical functions, called "wave functions" ( \psi , psi), to describe the electron's wave in 3D space, helping to define its probable location.

• Schrödinger Equation: H \psi = E \psi . This equation uses wave functions ( \psi ) to describe the electron's wave in 3D space.

• Solutions (Orbitals): Solving the Schrödinger equation yields multiple wave functions ( \psi ), each corresponding to a specific energy level. These electron wave functions are referred to as orbitals.

• Key Information from Orbitals: Orbitals define an electron's energy level and the shape of its wave.

• Probability and Electron Density: The square of the wave function ( \psi^2 ) gives the probability of finding an electron in a specific region of space (volume), also known as electron density. Denser areas in dot-density diagrams indicate higher electron density around the nucleus.

Characterization of Orbitals by Quantum Numbers

• Three quantum numbers are used to define the probability distribution of an electron in 3D space within an atom, encompassing its position, energy, shape, and size.

Principal Quantum Number ( n )

• Symbol: n .

• Also known as: Shell.

• Values: Positive integers: 1, 2, 3, 4, \ldots, \infty .

• Related to:

  • Size of the electron wave: Larger n means a larger orbital, indicating a higher probability of finding the electron further from the nucleus.

  • Energy of the electron wave/orbital: Larger n corresponds to higher energy orbitals.

  • All orbitals with the same principal quantum number belong to the same shell.

Secondary (Orbital Angular Momentum) Quantum Number ( l )

• Symbol: l .

• Also known as: Orbital angular momentum quantum number.

• Values: Integer numbers ranging from 0 to (n-1) for a given n .

• Related to: Defines the shape of the orbitals.

• Subshell Designation by Letters:

  • l=0 corresponds to an s subshell.

  • l=1 corresponds to a p subshell.

  • l=2 corresponds to a d subshell.

  • l=3 corresponds to an f subshell.

  • And so on (g, h, …).

• Number of Subshells: There are n subshells within each shell.

  • Example: For n=1 , l=0 (1s subshell). For n=2 , l=0, 1 (2s, 2p subshells).

• Grouping: Divides orbitals in a shell into smaller groups called subshells.

Magnetic Quantum Number ( m_l )

• Symbol: m_l .

• Values: Integer numbers ranging from -l to +l , including 0 , for a given l .

• Related to: Defines the orientation of orbitals relative to each other in space.

• Number of Orbitals: For each l value, there are (2l+1) possible m_l values, meaning there are (2l+1) orbitals in that subshell.

  • Example: If l=0 (s subshell), m_l=0 (1 orbital).

  • Example: If l=1 (p subshell), m_l = -1, 0, +1 (3 orbitals, typically p_x, p_y, p_z ).

  • Example: If l=2 (d subshell), m_l = -2, -1, 0, +1, +2 (5 orbitals).

• Grouping: Divides the subshells into individual orbitals.

Analogy for Quantum Numbers: Like an address to find where an electron lives:

• Province \rightarrow Principal Quantum # ( n ) (Shell)

• City \rightarrow Secondary Quantum # ( l ) (Subshell)

• Street \rightarrow Magnetic Quantum # ( m_l ) (Individual Orbital)

Shapes and Properties of Atomic Orbitals

s Orbitals

• Shape: All s orbitals are spherical.

  • The probability of finding the electron is uniform in all directions.

• Boundary Diagram: A spherical volume enclosing typically a 90% probability of finding the electron ( r_{90} ).

• Nodes: s orbitals have spherical nodes (or radial nodes), where the electron density is zero.

  • The number of spherical nodes for an s orbital is (n-1) .

  • 1s orbital: 0 nodes.

  • 2s orbital: 1 spherical node.

  • 3s orbital: 2 spherical nodes.

• Size and Energy: As n increases (e.g., from 1s to 2s to 3s), the orbital becomes larger and has higher energy.

p Orbitals

• Shape: p orbitals have two lobes.

  • High probability of finding electrons within these lobes, residing on opposite sides of the nucleus.

• Nodal Plane: A plane exists between the two lobes (at the nucleus) where the electron density is zero.

  • The phase of the wave changes across this nodal plane.

• Orientations: For a given n \ge 2 , there are three degenerate p orbitals, each oriented along a different Cartesian axis:

  • p_x : Lobes extend along the x-axis.

  • p_y : Lobes extend along the y-axis.

  • p_z : Lobes extend along the z-axis.

  • These three p orbitals are perpendicular to each other, possess similar shapes and sizes, and (in isolated atoms) have equal energies.

• Nodes: Higher n p orbitals (e.g., 3p, 4p) also include spherical nodes.

  • The total number of nodes (one nodal plane + spherical nodes) for any orbital is (n-1) . For 3p, this means (3-1) = 2 nodes (one nodal plane and one spherical node).

• Size and Energy: Higher n values correspond to larger and higher energy p orbitals (e.g., 3p orbitals are larger than 2p).

d Orbitals

• Shape and Orientation: For a given n \ge 3 , there are five degenerate d orbitals, each with distinct shapes and orientations.

  • Four of these have four lobes:

  • d_{xy} : Lobes lie in the xy-plane, positioned between the x and y axes.

  • d_{xz} : Lobes lie in the xz-plane, positioned between the x and z axes.

  • d_{yz} : Lobes lie in the yz-plane, positioned between the y and z axes.

  • d_{x^2-y^2} : Lobes lie in the xy-plane, positioned directly along the x and y axes.

  • These four orbitals have two nodal planes.

  • One orbital has a unique shape:

  • d_{z^2} : Features two lobes along the z-axis and a donut-shaped ring in the xy-plane.

  • This orbital has two nodal cones.

Orbitals inside the atom

• Orbitals are centered around the nucleus and are nested within each other, similar to layers (e.g., 1s, 2s, 2p, 3d orbitals filling progressively outward).

Summary of Atomic Models and Electron Behavior

• Bohr Model: Electrons are particles in fixed 2D orbits with defined energy ( n ).

• De Broglie Model: Electrons are also standing waves with a specific wavelength ( \lambda ).

• Schrödinger Model: Electrons are described by wave functions ( \psi ) in 3D orbitals, which have specific shapes and energy levels.

• Key Takeaways:

  • An electron's energy determines its orbital type (s, p, d, etc.).

  • Due to the uncertainty principle, we can only determine the probability ( \psi^2 ) of finding an electron in a given space (electron cloud), not its exact location and energy simultaneously.

  • Energy changes cause electrons to transition between different orbitals.

Applications and Practice Questions

Sketching Orbitals

• Task: Sketch the boundary diagrams for a 2s, 2pz, and 3px orbitals, paying attention to their size, shape, phase, and orientation.

  • 2s orbital: Draw a spherical shape with a visible inner spherical node (a barren region) and overall positive phase.

  • 2pz orbital: Draw two lobes along the z-axis, with a nodal plane in the xy-plane. One lobe typically has a positive phase, and the other a negative phase. It should be larger than 2s.

  • 3px orbital: Draw two lobes along the x-axis, with a nodal plane in the yz-plane, and one spherical node (within the lobes). Similar phase assignment as 2pz, but larger than 2pz.

• Task: Modify a sketch to show two perpendicular 2p_x and 2p_y orbitals.

  • Draw a 2p_x orbital along the x-axis and a 2p_y orbital along the y-axis, ensuring they are perpendicular and centered at the origin.

• Task: Modify a sketch to show a 3d_{x^2-y^2} orbital.

  • Draw four lobes lying in the xy-plane, positioned along the x and y axes.

Identifying Largest Orbitals and Quantum Numbers

• Question: Which orbital(s) is the largest: 2s, 4s, 3pz, 4pz?

  • Answer: 4s and 4pz are the largest. Orbitals with higher principal quantum numbers ( n ) are generally larger because the electron is more likely to be found further from the nucleus.

• Question: Write their sets of quantum numbers ( n, l, m_l ) for the orbitals listed.

  • 2s: ( n=2, l=0, m_l=0 )

  • 4s: ( n=4, l=0, m_l=0 )

  • 3pz: ( n=3, l=1, m_l=0 ) (The m_l=0 is conventionally assigned to the p_z orbital, or is one of the possibilities for p orbitals).

  • 4pz: ( n=4, l=1, m_l=0 ) (Similarly, m_l=0 for the p_z orbital).