General Chemistry (Chem-6A) Week 5: Wave Functions and Orbitals - Exhaustive Study Notes

PAGE 1: COURSE INFORMATION

• Course Title: General Chemistry (Chem-6A)
• Academic Term: Spring 2026
• Lecture Series: Week 5 - Wave functions and Orbitals
• Instructor: Lalit Deshmukh

PAGE 2: LIMITATIONS OF THE BOHR MODEL

• Historical Context: While Niels Bohr provided a revolutionary explanation for the hydrogen atomic spectrum, his model has significant shortcomings.
• Key Limitation: The Bohr model cannot be extended to atoms containing more than one electron ($+1$ electron systems).

PAGE 3: PIONEERS OF QUANTUM MECHANICS

• W. Heisenberg (1901-1976): Developed the Uncertainty Principle, a cornerstone of quantum theory.
• Erwin Schrödinger (1887-1961): Developed the wave equation and the wave mechanical model of the atom.

PAGE 4 & 5: HEISENBERG’S UNCERTAINTY PRINCIPLE

Fundamental Concept

• Observation (Mid-1920s): Heisenberg demonstrated that it is impossible to simultaneously determine both the exact position and the exact momentum of a subatomic particle (like an electron).
• Nature of Uncertainty: If an experimental measurement is performed to locate the exact position of an electron, that very act causes an inherent uncertainty in the electron's momentum.
• Causality: This uncertainty is not a byproduct of technical limitations in measuring equipment; rather, it is a fundamental property of the act of measurement itself.

Mathematical Formulation

• The Uncertainty Equation:$\Delta x \Delta p \geq \frac{h}{4\pi}$   - $\Delta x$ = Uncertainty in position ($x$).   - $\Delta p$ = Uncertainty in momentum ($p$).   - $h$ = Planck’s constant ($6.626 \times 10^{-34}\,J\cdot s$).
• Momentum and Velocity: Since momentum is defined as $p = mv$ and the mass ($m$) of the particle is typically known, the uncertainty in momentum ($\Delta p$) is driven by the uncertainty in velocity ($\Delta v$).

PAGE 6: PRACTICE PROBLEMS - UNCERTAINTY

• Practice Problem 1: Estimate the uncertainty in velocity of an electron if its position is known to within $5 \times 10^{-11}\,m$. (Given: mass of an electron $m = 9.11 \times 10^{-31}\,kg$, and $1\,J = 1\,kg\cdot m^2\cdot s^{-2}$).
• Practice Problem 2: Calculate the percent uncertainty if the electron is moving at $5 \times 10^6\,m/s$.
• Practice Problem 3: Consider a golf ball (mass = $46\,g$) traveling at $200\,km/h$. If the position is measured with a precision of $1\,mm$, calculate the uncertainty in its speed.

PAGE 7: THE QUANTUM MECHANICAL MODEL OF THE ATOM

• The Schrödinger Model:   - Developed to explain atomic results for all atoms, not just hydrogen.   - Electron Behavior: The model assumes electrons behave as waves and utilizes a wave equation.   - Orbitals: It describes 3-D regions of space where there is a high probability of finding an electron.   - Probability vs. Determinism: Instead of defining fixed paths (orbits), the model discusses regions of probability.

PAGE 8 & 9: SCHRÖDINGER’S WAVE EQUATION

The Wave Function (\Psi)

• Purpose: The equation describes the probability of finding an electron in a specific region.
• Variables: The wave function is defined as $\Psi = \Psi(x, y, z)$.
• Probability Density ($\Psi^2$): The square of the wave function represents the probability density of finding an electron at a specific point in space.

Defining Orbitals

• Representation: Regions of high probability are called orbitals.
• Visual Density: In dot-density diagrams, the density of dots is directly proportional to the probability of finding an electron at that location.
• 99% Boundary: Typical orbital diagrams (like spheres) enclose a volume where there is a 99% probability of finding the electron.
• Energy Restrictions: Rather than restricting the electron to a physical path, wave functions restrict the allowed energy of the system.

PAGE 10: ORGANIZATION OF THE ELECTRON CLOUD

• Electron Cloud: The entire space outside the nucleus where electrons reside.
• Hierarchical Structure:   1. Shells (n): The primary division of the electron cloud.   2. Subshells (\ell): Divisions within shells.   3. Orbitals: Divisions within subshells; these are the actual 3D regions.
• Capacity: Every single orbital can contain a maximum of 2 electrons.

PAGE 11-13: THE FOUR QUANTUM NUMBERS

Historical Context

• Bohr Model: Utilized only one quantum number ($n$) to describe the distance of the electron from the nucleus ($n=1, 2, 3, 4, …$).
• Schrödinger Model: Uses four quantum numbers to completely define the state of an electron.

Quantum Number Details (Table 1D.2)

1. Principal Quantum Number ($n$):    - Values: $1, 2, 3, …$    - Specifies: The shell.    - Indicates: Size and (indirectly) energy. Larger $n$ implies a larger and more energetic orbital.
2. Orbital Angular Momentum / Azimuthal Quantum Number ($\ell$):    - Values: $0, 1, …, n-1$    - Specifies: The subshell.    - Indicates: Shape of the orbital.    - Subshell Designations:      - $\ell = 0$: $s$ (Sharp)      - $\ell = 1$: $p$ (Principal)      - $\ell = 2$: $d$ (Diffuse)      - $\ell = 3$: $f$ (Fundamental)
3. Magnetic Quantum Number ($m_\ell$):    - Values: $-\ell, …, 0, …, +\ell$    - Specifies: Orbitals of the subshell.    - Indicates: Spatial orientation.
4. Spin Magnetic Quantum Number ($m_s$):    - Values: $+\frac{1}{2}, -\frac{1}{2}$    - Specifies: Spin state.    - Indicates: Direction of electron spin.

PAGE 16-18: ALLOWED VALUES AND DESIGNATIONS

• Orbital Notation: Orbitals are named by the numerical value of $n$ followed by the letter code for $\ell$ (e.g., $1s, 2p, 3d$).
• Practice Question: Explain why there is no $1p$ orbital. (Hint: Look at the rule for allowed $\ell$ values).

Table 5.2: Hierarchy of States (n=1 to n=4)

• n=1: $\ell=0 (1s)$; $m_\ell=0$; Total = 1 orbital.
• n=2:   - $\ell=0 (2s)$; $m_\ell=0$; Total = 1 orbital.   - $\ell=1 (2p)$; $m_\ell=1, 0, -1$; Total = 3 orbitals.
• n=3:   - $\ell=0 (3s)$; $m_\ell=0$; Total = 1 orbital.   - $\ell=1 (3p)$; $m_\ell=1, 0, -1$; Total = 3 orbitals.   - $\ell=2 (3d)$; $m_\ell=2, 1, 0, -1, -2$; Total = 5 orbitals.
• n=4:   - $\ell=0, 1, 2$ (as above).   - $\ell=3 (4f)$; $m_\ell=3, 2, 1, 0, -1, -2, -3$; Total = 7 orbitals.

PAGE 19, 20, 34, 35: ORBITAL SHAPES AND PRACTICE

• p Orbitals ($\ell=1$): Dumbbell-shaped, oriented along the axes ($p_x, p_y, p_z$). They possess a nodal plane where the probability is zero.
• d Orbitals ($\ell=2$): There are five 3d orbitals: $d_{x^2-y^2}, d_{z^2}, d_{xy}, d_{xz}, d_{yz}$.
• Practice Problem: Write the values of $n, \ell,$ and $m$ for a shell where $n = 5$.

PAGE 21-23: ELECTRON SPIN AND THE STERN-GERLACH EXPERIMENT

The Pauli Exclusion Principle

• Definition: No two electrons in the same atom can have the exact same set of four quantum numbers ($n, \ell, m_\ell, m_s$).

Experimental Verification (1922)

• The Stern-Gerlach Experiment: Performed by Otto Stern and Walther Gerlach using a beam of silver atoms passed through a magnetic field.
• Result: The beam split into two distinct spots on a collection plate, rather than a continuous smear. This provided the experimental proof of directional quantization (electron spin).

PAGE 24-27: SUMMARY OF QUANTUM QUANTITIES

Capacity per Level

• n=1: 1 subshell ($1s$), 1 orbital, 2 electrons.
• n=2: 2 subshells ($2s, 2p$), 4 orbitals ($1+3$), 8 electrons.
• n=3: 3 subshells ($3s, 3p, 3d$), 9 orbitals ($1+3+5$), 18 electrons.
• n=4: 4 subshells ($4s, 4p, 4d, 4f$), 16 orbitals ($1+3+5+7$), 32 electrons.

Quantum Constraints Recap

• $n \neq 0$; Must be $1, 2, 3, …$
• $\ell$ can be any integer from $0$ to $n-1$.
• $m_\ell$ can be any integer from $-\ell$ to $+\ell$.
• $m_s$ is strictly $+\frac{1}{2}$ or $-\frac{1}{2}$.

PAGE 28: PRACTICE - ELECTRON CAPACITY

What is the maximum number of electrons specified by:

• a. $n = 3$ (Total = 18)
• b. $n = 3, \ell = 1$ (Total = 6)
• c. $n = 3, \ell = 1, m_\ell = 0$ (Total = 2)
• d. $n = 3, \ell = 1, m_\ell = 0, m_s = \frac{1}{2}$ (Total = 1)

PAGE 29-31: THE PERIODIC TABLE CONNECTION

• s-block: Groups 1A, 2A, and Helium.
• p-block: Groups 3A to 8A (excluding He).
• d-block: Transition metals.
• f-block: Lanthanides and Actinides.
• Comparison: Unlike the Bohr model that uses only $n$, the Quantum Mechanical model matches the periodic table's structure through subshells ($\ell$) and orbital orientations ($m_\ell$).

PAGE 32-33: ORBITAL ENERGIES AND STABILITY

Factors Affecting Energy:

1. Orbital Shape: Likelihood of finding electrons close to the nucleus.
2. Nuclear Charge: Electrons are stabilized by Coulombic interaction with protons.
3. Shielding: Repulsion from other electrons decreases Coulombic stabilization.

• Stability Rule: Electrons closer to the nucleus are more stable; hence, inner shells are lower in energy. Energy increases as $n$ increases.

PAGE 36-37: NODAL SURFACES

• Definition: Surfaces where electron probability density is zero.
• Energy Correlation: More nodes lead to higher energy.
• Radial Nodes: Spherical nodes located between shells. Number of radial nodes = $n - 1$.
• Angular Nodes: Planar nodes. Number of angular nodes = $\ell$.
• Comparison (Within a shell): Electrons in $s$ orbitals ($\ell=0$) are more likely to be close to the nucleus than $p, d, \text{ or } f$, making $s$ orbitals the most stable within a shell. Energy order: s < p < d < f.

PAGE 39-41: NUCLEAR CHARGE AND SHIELDING

Coulombic Attraction

• Formula: $F = k\frac{Ze}{r^2}$.
• Impact: Higher nuclear charge ($Z = \text{protons}$) leads to more stable orbitals (e.g., Carbon at $+6$ is more stable than Beryllium at $+4$).

Shielding Mechanics

• Effective Nuclear Charge ($Z_{eff}$): Shielding reduces the full nuclear charge seen by an electron.
• Degree of Shielding:   - Inner Electrons: Provide the greatest shielding to outer electrons.   - Same-Orbital Electrons: Shield each other slightly.
• Consequence: Outer orbitals (higher $n$) are less stable and easier to remove.

PAGE 42-45: ELECTRON CONFIGURATION RULES

Theoretical Order

• Degenerate Orbitals: Orbitals with the same $n$ and $\ell$ (e.g., the three $2p$ orbitals) have equal energy.
• The Energy Sequence: 1s < 2s < 2p < 3s < 3p < 4s < 3d < … (Note that energy of $4s$ is less than $3d$).

The Three Rules of Filling

1. Aufbau Principle: Electrons fill the lowest available energy levels first.
2. Pauli Exclusion Principle: Max 2 electrons per orbital; they must have opposite spins.
3. Hund’s Rule: Within a subshell, electrons fill orbitals singly with parallel spins before they begin to pair up.

PAGE 46-53: ORBITAL DIAGRAMS AND PRACTICE

• Visuals: Lines/boxes represent orbitals; arrows represent electrons.
• Subshell Orbital Counts:   - $s$ = 1 blank   - $p$ = 3 blanks   - $d$ = 5 blanks   - $f$ = 7 blanks
• Nitrogen Practice: 1s, 2s, 2p (with three unpaired arrows in 2p per Hund's Rule).
• Germanium Practice: Locating Ge on the periodic table and filling up to the $4p$ subshell.
• Ionic Configurations:   - $O^{2-}: 1s^2 2s^2 2p^6$ (Isoelectric with Neon).   - $F^{-}: 1s^2 2s^2 2p^6$.

PAGE 54-58: SHORTHAND AND VALENCE ELECTRONS

• Shorthand Method: Replace core configurations with noble gas symbols in brackets (e.g., Carbon is $[He]2s^2 2p^2$).
• Valence Electrons: Electrons in the outermost shell ($n_{max}$) of an atom. They determine chemical reactivity.
• Examples:   - Carbon: 4 valence electrons ($2s, 2p$).   - Sodium: 1 valence electron ($3s$).   - Chlorine: 7 valence electrons ($3s, 3p$).
• Note: For ions like $O^{2-}$, count valence based on the final filled configuration.

PAGE 59-60: EXCEPTIONS TO FILLING RULES

• Unusual Configurations: Some elements stabilize by moving an $s$ electron to the $d$ subshell to achieve a half-filled or fully-filled state.
• Key Exceptions:   - Cr (Chromium): $[Ar] 4s^1 3d^5$   - Cu (Copper): $[Ar] 4s^1 3d^{10}$   - Ag (Silver): $[Kr] 5s^1 4d^{10}$   - Au (Gold): $[Xe] 6s^1 5d^{10}$

PAGE 61-62: ELECTRON CONFIGURATION TERMINOLOGY

• Isoelectric: Atoms/ions with identical electron configurations (same number of electrons).
• Unpaired Electron: A single electron alone in an orbital.
• Paired Electron: Two electrons occupying the same orbital.
• Paramagnetic: A substance attracted to a magnetic field due to the presence of unpaired electrons.
• Diamagnetic: A substance NOT attracted to a magnetic field because all electrons are paired.