Ch 6 Lecture 3
Wave Mechanical Model of the Atom
Introduction to Quantum Mechanics
Quantum mechanics is an essential topic for understanding atomic theory in chemistry.
The wave mechanical model of the atom is the currently accepted model.
Wave-Particle Duality
Matter exhibits both wave-like and particle-like behavior, a concept known as wave-particle duality.
Schrödinger Equation and Wave Function
Schrödinger Equation
The Schrödinger equation provides solutions that describe the wave function, denoted by the Greek letter Ψ (psi).
The wave function represents a matter wave and evolves over time, typically expressed in an exponential decay format: ext{s} = a imes e^{-ikx}
Where:
a is a constant representing the initial amplitude.
i is the imaginary unit (non-physical).
k is a constant termed the wave number (represents the stiffness of the wave).
x is a position along the x-axis.
Interpretation of Wave Function
The wave function decays over space and time; however, its specific values do not need immediate understanding for students.
When graphing ext{Ψ} , the results reflect an exponential decay shape.
Born Rule
Born Rule (Max Born): To interpret the wave function physically, we square it to eliminate the imaginary component:
|{Ψ}|^2 gives the probability of locating the particle in space
This indicates that if representing an electron, it cannot be localized precisely anywhere due to equal probability distribution throughout space.
Implications for Electrons
There is no specific position of electrons in an atom; rather, electrons are best described as fields permeating all space.
Pauli Exclusion Principle
Wolfgang Pauli
In the 1920s, Wolfgang Pauli introduced the Pauli Exclusion Principle.
This principle is crucial for understanding electron configurations in atoms.
Definition of the Pauli Exclusion Principle
No two electrons in an atom can have identical quantum numbers, an idea essential when constructing electron configurations.
Example with Electrons in an Atom
Illustration of two electrons in an atom:
Electron 1: Wave function ext{Ψ}_1 in state α.
Electron 2: Wave function ext{Ψ}_2 in state β.
Pauli derived his principle noting the unique states of electrons represented mathematically as:
ext{S} = ext{Ψ}1 (α) imes ext{Ψ}2 (β) - ext{Ψ}1 (β) imes ext{Ψ}2 (α)
Interpretation of Pauli's Equation
This determinant-like equation signifies which configurations are possible (
true statements) versus impossible (false statements).The negative sign in the equation designates that those configurations are not allowed.
Consequences of Identical States
Setting the states α and β equal leads to total wave function ext{S} = 0 :
Probability collapses (total probability = 0) means electrons cease to exist, reaffirming the need for distinct quantum numbers.
Quantum Numbers
Four Key Quantum Numbers
Principal Quantum Number (n): Corresponds to the energy level and size of the atomic orbital; values range from 1 to 7 in the periodic table.
Proportional to energy content.
Angular Momentum Quantum Number (l): Articulates the shape of the orbital and is associated with subshells:
Values: 0 (s), 1 (p), 2 (d), 3 (f).
Magnetic Quantum Number (m_l): Indicates the orientation of the orbital; ranges from -l to +l (total of 2l + 1 values).
Spin Quantum Number (m_s): Represents the intrinsic spin of the electron; values are +1/2 (spin up) and -1/2 (spin down).
Differences in Fermions and Bosons
Electrons are categorized as fermions, adhering to the Pauli exclusion principle, in contrast to bosons, which do not.
Structure of the Periodic Table
Atomic Structure Encoded in the Table
The periodic table reflects the underlying quantum mechanical structure:
S and P blocks correspond to electrons in the configurations where their respective quantum numbers are defined.
Electron Distribution in Atomic Orbitals
The periodic table not only lists the elements but also visualizes the quantum states and distribution:
S: 2 electrons (2 groups), P: 6 (6 groups), D: 10 (10 groups), F: 14 (7 groups).
Visual Representation of Orbitals
Drawing Orbitals
Students should practice properly sketching and labeling different orbital shapes:
S Orbitals: Spherical (one orientation).
P Orbitals: Figure-eight shapes, with three orientations along axes (px, py, p_z).
D and F Orbitals: More complex shapes (noting that detailed drawing of F orbitals is not generally required).
Quantum Angular Momentum
Total orbital angular momentum is represented as follows:
Total Orbital Angular Momentum Formula
L = ext{sqrt}[l(l + 1)] ext{ ℏ} where ℏ is the reduced Planck's constant.
Angular Momentum in S and D Orbitals
S: Zero angular momentum (does not interact with the nucleus).
D: Non-zero angular momentum, influencing electron location related to nuclear interactions.
Heisenberg's Uncertainty Principle
Concept Overview
The Heisenberg uncertainty principle relates position and momentum, mathematically expressed via: ext{ΔxΔp} ext{ ≥ } rac{ ext{ℏ}}{2}
Indicates the intrinsic limits of what can be known about a particle's position and momentum simultaneously.
Implications for Atomic Structure
For s electrons, due to the principle, tunneling occurs allowing escape from the nucleus despite being confined.
Example of Computing Electrons’ Behavior
If a s electron were inside a nucleus, physics dictates it would need significant energy to remain confined, leading to an impossible scenario where the speed surpasses light.
Quantum Number Examples
Lithium Example
Lithium (Z=3) configuration: 1s² 2s¹
Electron assignments:
Electron 1: 1s, l=0, ml=0, ms=+1/2
Electron 2: 1s, l=0, ml=0, ms=-1/2
Electron 3: 2s, l=0, ml=0, ms=+1/2
Summary of Quantum Number Values
n: 1 (for first two) and 2 (for third electron)
l: 0 for all electrons in s orbitals.
m_l: 0 for s electrons.
m_s: Mix of +1/2 and -1/2 for spins.
Conclusion
The lecture concludes by emphasizing the complexity of electron arrangements, quantum state implications, and the profound link among quantum mechanics, atomic behavior, and the structure of the periodic table.