Module 2.3 Development of Quantum Theory 1

Learning Objectives

Understand the general idea of the quantum mechanical description of electrons in an atom and orbitals
List and describe traits of the four quantum numbers that form the basis for completely specifying the state of an electron in an atom

Previously: Bohr model only worked on hydrogen. Electrons were said to be at fixed distances only and defined by a single quantum number n. Scientists needed something that could extend Bohr’s model to other elements.

Who took wave-particle duality and changed it to [this] equation?
- Schrödinger
When Schrödinger applied his equation to atoms what happened?
- He was able to extend Bohr’s model to ALL atoms
What else can the Schrödinger equation be used for
- Give three-dimensional map of where electron supposed to be located
- What’s this map called?
  - Atomic Orbital
What does this orbital map allow us to determine
- distribution of electron’s density w/respect to nucleus
- What does this portend?
  - It is the foundation of quantum mechanics

UNDERSTANDING QUANTUM THEORY OF ELECTRONS IN ATOMS

The goal of this section is to understand electron orbitals (locations), energies & properties. This is Quantum Theory.

What does quantized energy mean?
- Electrons can only exist in discrete energy levels
What can quantized energy do?
- Equal to only specific values
- Can jump from one energy level (abruptly)
- Can’t stay between levels
What is n?
- The principle (first) quantum number
  - what does it do
    - Define the location of energy level
- same n in Bohr
What is another name for principle quantum number
- Shell Number
  - What is a shell number
Describe shell numbers
- Concentric spheres radiating from the nucleus
- electrons likely found there
  - Is it within the sphere or the boundary or both?
    - They are mutually exclusive

What happens when the atom moves further from the nucleus?
- Energy increases
- Higher shell number
How do protons stabilize orbitals
- Using electrostatic attraction from their positive charge & negative charge of e-
- Therefore the farther the e- the greater the E
What happens when an e- transition to a higher energy level?
- E becomes absorbed (+E)
- the opposite happens when it is released (-E)
  
  Equation for transition to a different energy level, where k is a defined constant, and nf is final orbital and ni is initial
So what does the principle quantum number describe?
- Size and energy of an orbital
How many quantum numbers are there?
- Three:
  - n, l, m, d
What is the secondary quantum number?
- l
- what does it characterize?
  - angular momentum
What is the highest value of of l
- n - 1 (one less than the principle quantum number)
What can n=1 have a value of?
- One l value
  - l = 0 (only this one value)
- n = 2?
  - l = 0 and l = 1
So if the principle quantum number specifies ___ of orbital, what does the secondary quantum number do?
- n signifies size and E
- l signifies angular momentum (shape)
What do orbitals w/same value of l called?
- Subshell
  - Of what, specifically though?
    - p subshell
    - It is equivalent to l = 1 (corresponds to p orbitals, but for l=1 only)
How does n orbitals equate to a certain subshell?
- n = x has a corresponding p subshell
  - If n = 3, p = 3, etc
What are the orbitals of l = 2
- *d orbitals n, l, m, d
- What orbitals follows d orbitals?
  - f-, g-, h-
  - For l = 3, 4, 5
What do the s, p, d, f electron density distribution look like?
- S = sphere
- p = dumbell
- d and f are much more complex:
What is the magnetic quantum number?
- It is a quantum number, ml, that specifies the relative spatial orientation of a particular orbital
- ml can equal all integers from -l to l
What are the total numbers of possible orbitals w/same value of l (same subshell)?
- One s-orbital in s subshell (l=0)
- look at the graphic above to see how many subshells each orbital has!
Looking at the graph above, what is the principle quantum number?
- The integer in front (2 from 2s, 3 from 3p, etc) from the principle quantum number n
- what defines the subshell, then?
  - The letter
  - Are there more possibilities for orbitals?
    - Yes, for l = 0 for s orbitals, 1 for p orbitals, 2 for d orbitals
    - Finally, there are more than one orbital for l greater or equal to 1, each corresponding to specific value of ml
Orbitals of the same subshell are called?
- Degenerate (same energy)
  - For example, 2p has three degeneracy
How many orbitals in 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p?
- See the graph below

-The above graphic is taken from Section 2.4

What is the fourth quantum number?
- Spin quantum number (ms)
What were the other 3 quantum numbers again?
- n, l, ml
- These are properties of specific atomic orbitals that define where electrons can be found at any given point in time
  - Orbitals were discovered by Schrödinger
What is an electron spin?
- it is a completely quantum phenomenon with no analogues in the classical realm, it is completely quantum mechanics only
  - Cannot be derived from Schrödinger
  - Not related to normal spatial coordinates (Cartesian, etc)
- It is simply a spin rotation intrinsic to electrons
How many values can the electron magnitude have?
- Only one value
How many spins can an electron go?
- Two quantized states
  - α state
    - Z component in positive z axis
      - Spin quantum number
  - β state , w/ z component in negative z axis
  - Any electron, regardless of orbital, can only one of these two quantized values of the spin quantum number
  - The energies of the two quantum spins (using the two equations above) are different if an external magnetic field is applied
What equations show a negative spin? A positive one?
- - Negative spin
- - Positive spin
- both represented as the spin quantum number, ms

THE PAULI EXCLUSION PRINCIPLE

What does it ultimately mean?
- No more than Two electrons in the same atom can have the same set of four quantum numbers
  - Therefore any electronic orbital can only be populated by
    - Zero, one, or two Electrons for each degenerate subshell of an orbital
- What are the four quantum numbers again
  - n, l, ml, ms