Honors Chemistry
Chapter 7: Quantum Mechanics
7.1 Wave Properties
• Wavelength (l) =
distance between
two in-phase points
• Measured in meters
• Frequency (n) =
number of waves per second
• Measured in Hertz (Hz)
• Amplitude (y) = distance of maximum
displacement from rest position
• Amplitude corresponds to wave energy
7.1 The Wave Equation
• v = ln
• Find the wavelength of a 256 HZ (middle
C) sound wave traveling at 343 m/s.
• v = ln
• 343 m/s = l (256 Hz)
• l = 1.34 m
• Try this….
• Find the frequency of a 25.0 cm wave traveling at
0.75 m/s.
7.1 Electromagnetic Radiation
•
•
•
•
•
•
•
James Clerk Maxwell (1873)
Mathematical description of light waves
Light is an electromagnetic wave
Speed of light (c) is constant
c = 2.99792458 x 108 m/s
To 3 sig dig, 3.00 x 108 m/s is fine
Try this….
• Find the frequency of a 250 nm light wave. (Don’t
forget about the “nano” prefix!)
7.1 Electromagnetic Spectrum
Radio, micro, IR, ROYGBIV, UV, X, g
long l ------------------------------- short l
low n -------------------------------- high n
Radio wave end of the spectrum is low
energy radiation
• Gamma ray end is high energy radiation
• Black body radiation
• Wave theory fails to account for this!
•
•
•
•
7.1 Quantum Theory
• Max Planck (1900)
• Energy is emitted and absorbed only in
small, discrete packets called quanta
• Energy of a quantum of energy given by
E = hn
• h = 6.626 x 10-34 Js (Planck’s constant)
• Correctly accounts for blackbody curves
• Planck has no idea why it works!
7.1 Quantum Theory
• Find the energy of a 2.50 x 1014 Hz light
wave.
• E = hn
• E = (6.626 x 10-34 Js)(2.50 x 1014 Hz)
• E = 1.66 x 10-19 J
• A quantum holds a tiny amount of energy!
• Try this….
• Find the energy of a 475 nm light wave.
• Hint: Use the wave equation first!
7.2 The Photoelectric Effect
• Albert Einstein (1905)
• Electrons ejected from surface of
metal exposed to light
• Depends on frequency of light
• Electrons ejected at a certain cutoff frequency
• Above cutoff n, electrons leave with more energy
• Bright light ejects more electrons
• Quantum theory explains results
• Light is made of quanta called photons
7.3 Spectroscopy
• Emission spectra – light given off by
glowing objects
• Can be continuous or discontinuous
• Line spectra – series of bright lines
emitted by gas phase atoms
• Pattern of bright lines is characteristic of
the element that is glowing
• Absorption spectra – dark lines in
spectrum as light passes through a gas
7.3 Bohr’s Model
• Niels Bohr (1913)
• Electron energies are quantized
• Only certain orbits are allowed
•
- RH
En = -----n2
• RH = 2.18 x 10-18 J (Rydberg constant)
• n = 1, 2, 3, 4, ….
7.3 Bohr’s Model
• DE = Ef – E0
•
-RH -RH
DE = ----- - ----nf2
n02
• Factor out RH
•
1
1
DE = RH (----- - ----- )
n02
nf2
Link to Hydrogen energy states
7.3 Bohr’s Model
• Find the energy of a photon of light
emitted by an electron jumping from level
5 down to level 2.
• DE = RH (1/n52 – 1/n22)
• DE = (2.18 x 10-18 J)(1/25 – 1/4)
• DE = -4.58 x 10-19 J
• Try this….
• Find the energy of the jump from level 1 to level 4.
• Find the frequency of the light produced.
7.4 Duality
• Louis de Broglie (1924)
• Electrons can be treated as waves
• Each orbit must contain a whole number of
waves…explains orbit quantization!
•
h
l = ---mv
• mv is momentum (p), so we can write l = h/p
• Verified by Davisson, Germer, and Thomson
Link to quantum atom model
7.4 Duality
• Find the wavelength of a 3.00 kg duck
flying at 5.00 m/s.
• l = h/mv
• l = (6.626 x 10-34 Js) / (3.00 kg)(5.00 m/s)
• l = 4.42 x 10-35 m
• Try this….
• Find the wavelength of an electron traveling at
500,000 m/s. (me = 9.11 x 10-31 kg)
7.5 Uncertainty Principle
• Werner Heisenberg (1926)
• Complementary variables cannot
be known to arbitrary precision
• dp dq ≥ ħ/2
• Minimum limits to uncertainties in values
are inversely proportional
• Position and momentum are an important
complementary pair
• dx dpx ≥ ħ/2
7.5 Uncertainty Principle
• Find the uncertainty in velocity of an electron
confined to a hydrogen atom
(dx = 0.037 nm).
• dx dpx ≥ ħ/2
• (3.7 x 10-11 m) dp ≥ 5.27 x 10-35 Js
• dp ≥ 1.4 x 10-24 kg m/s
• p = mv
• 1.4 x 10-24 kg m/s = (9.11 x 10-31 kg) dv
• dv = 1.5 x 106 m/s
7.5 Uncertainty Principle
• Try this…
• Find the uncertainty in position of a 20.0 mg fly
whose position is known to within ±0.5 mm.
• Uncertainty limits are not significant for
macroscopic objects, but they are
significant to subatomic particles
• Cannot know the position and momentum
of an electron at the same time!
• Concept of orbits will not work
7.5 Quantum Mechanics
• Erwin Schrödinger (1926)
• Schrödinger equation – treat electron as a
standing wave surrounding the nucleus
• Schrödinger equation is ugly!
• Solve for amplitude function (y)
• Remember – amplitude is energy
• Produces an energy diagram like
Bohr’s, but this one actually works
• Wave function has no physical meaning
7.5 Copenhagen Interpretation
•
•
•
•
•
•
•
•
•
Max Born (1926)
y2 denotes probability
Electron is delocalized
Wave function collapses on observation
electron density refers to magnitude of the
probability wave for the electron
Orbital = spatial probability distribution
Electron clouds
Objections: Schrödinger’s Cat
Other interpretations
7.6 Quantum Numbers
• Set of numbers that describe the
distribution of electrons in the atom
• Principal Quantum Number (n)
• n = 1, 2, 3, 4, …
• Corresponds to the n value used by Bohr
• Describes energy level of the shell
• Defines the size of the electron cloud
7.6 Quantum Numbers
• Angular Momentum Quantum Number (l)
• l = 0, 1, 2, … , n – 1
• There are a total of n values
•
•
•
•
Sublevels of the energy level
Angular distribution of electron cloud
For hydrogen, sublevels are degenerate
Correspond to fine structure spectral lines
• l = 0 is s orbital
• l = 1 is p orbital
l = 2 is d orbital
l = 3 is f orbital
7.6 Quantum Numbers
• Magnetic Quantum Number (ml)
• ml = -l, … , 0, …, +l
• There are a total of 2l + 1 values
•
•
•
•
Number of degenerate orbitals in sublevel
Spatial orientation of the orbital
Zeeman Effect
Electron Spin Quantum Number (ms)
• ms = +½, -½
• Two possible electron spin states
• Spin up, spin down
7.7 Atomic Orbitals
• s, p, d, f orbitals
• Radial probability distributions
• distance from nucleus of high e- probability
7.7 Atomic Orbitals
• Angular probability distributions
• Show regions of high e- probability
• Cool 3d pictures
7.7 Orbital Energies
• n determines energy
• For H, all subshells are degenerate
• Multielectron atoms
each subshell lies
at a different energy
• Shielding effect
• Fill lowest energy
orbitals first
7.7 The Diagonal Rule
• Rule of thumb
• Shows the order
in which orbitals
are filled
• Paramagnetic
• Unpaired e• Attracted to mag
• Diamagnetic
• Paired e• Not attracted
7.8 Electron Configurations
•
•
•
•
•
•
•
•
H has 1 electron
Put it in 1s
Write it 1s1
Read “one-s-one”
What about He?
You got it…1s2
Keep filling 1s until it is full
But when is it full?
7.8 Pauli Exclusion Principle
• No two electrons may share the exact set
of quantum numbers
• Consider Helium’s 1s2 configuration
• First electron: n = 1, l = 0, ml = 0, ms = +½
• Second electron: same n, l, ml
• ms must be -½
• No room for more electrons in 1s orbital
• Each orbital can hold only two electrons!
7.8 Electron Configurations
•
•
•
•
•
•
What is the electron configuration of Li?
1s2 2s1
What about N?
1s2 2s2 2p3
But how are p electrons organized?
Hund’s Rule – arrange electrons in such a
way as to maximize total spin state
• Put e-’s in separate orbitals, same spin
7.9 Aufbau Principle
•
•
•
•
•
•
Build up on previous e- configurations
Each atom adds one more eExpress configuration with noble gas core
Al = 1s2 2s2 2p6 3s2 3p1
First 3 terms are the same as Ne config.
Write it as [Ne] 3s2 3p1
7.9 Exceptions
• Particularly stable configurations
• Full sublevel
• Half-full sublevel
• Some transition metals rearrange
• Cr = [Ar] 4s2 3d4
• Just missed the stable half-full 3d5
• Kick one e- up to d to get [Ar] 4s1 3d5
• What other family would do this?
• Cu = [Ar] 4s2 3d9  Ar 4s1 3d10
7.9 Periodicity of Electron
Configuration