1. Chapter 7: The Quantum-Mechanical Model of the Atom (7.4-7.5)
By: Gagana Yaskhi

2. Quantum-Mechanical Model
Quantum-mechanical model was first proposed by Louis de Broglie
At the time, electrons were thought to have wave nature
If an electron beam passes through 2 slits, an interference pattern occur but it is not caused by pairs of electrons interfering with each other, it is caused by single electrons interfering with themselves
The wave nature of the electron is an inherent property of individual electrons

3. De Broglie Relation
Single electron traveling through space has wave nature and its wavelength is related to kinetic energy
shorter wavelengths move faster and have higher kinetic energy
de Broglie relation: λ= h/mv
λ is wavelength
h is Planck’s constant (6.626 x 1023 J·s)
m is mass of electron
v is velocity

4. Practice Problem
Calculate the wavelength of an electron traveling with a speed of 2.65 x 106 m/s. (Hint: mass of electron=9.11 x kg)
λ= h/mv= (6.626 x 1023 J·s)/(9.11 x kg)(2.65x 106 m/s)
λ= 2.74 x m

5. Uncertainty Principle
Electron diffraction experiment, that was performed to show which slit an electron travels through by using a laser beam directly behind slits, determined than we can never both see the interference pattern and simultaneously determine which hole the electron goes through
We can’t observe the wave and particle nature of the electron at the same time
Wave and particle nature are complementary properties (exclude one another; the more we know of one, the less we know of the other)
Velocity of electron= wave nature
Position= particle nature

6. Uncertainty Principle
Heisenberg’s uncertainty principle: Δ x x mΔ v ≥ (h/4 π)
Δ x is uncertainty of position
Δ v is uncertainty of velocity
m is mass
The more accurately you know position of electrons (smaller Δ x) the less accurately you know velocity (bigger Δ v)
An electron is observed as either a particle or wave but not both at once

7. Indeterminacy and Probability of Distribution Map
Particles move in a trajectory (path) determined by the particle’s velocity, position, and forces acting on it according to Newton’s laws of motion
Position and velocity are required to predict trajectory so we cannot know the trajectory of electrons
Newton’s laws of motion are deterministic (the present determines the future)
In quantum mechanics, probability distribution maps statistically show where an electron is most likely to land under given conditions as we cannot know its exact trajectory
Indeterminacy describes the behavior of an electron because future path of an electron cannot be determined

8. Quantum Mechanics and the Atom
Since velocity is directly related to energy, position and energy are complementary properties
Orbital: probability distribution map showing where an electron is most likely to be found
Spatial distribution of electron is important to bonding
Mathematical derivation of energies and orbitals for electrons comes from Schrödinger’s equation:
Wavefunction describes wavelike nature of electrons
H= total energy(kinetic and potential) of the electron in at atom
E= actual energy of electron
Ψ2 represents an orbital

9. Quantum Mechanics and the Atom
Each orbital is specified by 3 quantum numbers:
n= principal quantum number
integer that determines overall size and energy of orbital: n=1,2,3…
For hydrogen atoms, energy with quantum number n (En) = -2.18x10-18 J (1/n2) (n= 1,2,3,…)
l= angular momentum quantum number
Integer that determines shape of orbital: l=0,1,2…(n-1)
ml= magnetic quantum number
Integer that specifies orientation of orbital: m= +l to -l

10. Quantum Mechanics and the Atom
A fourth number ms= spin quantum number specifies orientation of spin of electron
Electrons can spin up (ms=+½) or spin down (ms=-½)
Each combination of n,l, and ml specifies an atomic orbital
n=1, l=0, ml=0 is 1s orbital
Orbitals with same n value are in the same principal levels or principal shell
Orbitals with same n and l are in the same sublevel (subshell)
# of sublevels = n
# of orbitals in any sublevel= 2l + 1
# of orbitals in a level= n2

11. Practice Problem
What are the quantum numbers and names (for example, 2s, 2p) of the orbitals in the n=4 principal level? How many n=4 orbitals exist?
n=4; therefore, l= 0,1,2,and 3
16 orbitals total l
Possible ml values
Orbital name
4s (1 orbital) 1
-1,0,+1
4p (3 orbitals) 2
-2,-1,0, +1, +2
4d (5 orbitals) 3
-3, -2, -1, 0, +1, +2, +3
4f (7 orbitals)

12. Atom Spectroscopy Explained
When an atom absorbs energy, an electron in a lower energy level is promoted to a higher one
The new configuration is unstable and the electron falls back to the lower energy orbital and releases a photon of light equal to the energy difference between the two levels
Δ E= Efinal - Einitial = x J (⅟nf2 - ⅟ni2)
Δ Eatom = -Ephoton
Transition between orbitals further apart in energy produces light in higher energy and shorter wavelength

13. Practice Problem
Determine the wavelength of light emitted when an electron in a hydrogen atom makes a transition from an orbital in n=6 to an orbital in n=5.
Δ Eatom= E5 - E6
= x J (⅟52 - ⅟62)
= x J
Δ Eatom = -Ephoton= x J
λ= hc/E
=(6.626 x 1023 J·s)(3.00 x 108 m/s)/ (2.664 x J)
= 7.46 x 10-6 m