Download Chapter 7: The Quantum Mechanical Model of the Atom I. The

Chapter 7: The Quantum Mechanical Model of the Atom
I. 
The Nature of Light: Its Wave-like Nature
A.  Light is a form of electromagnetic radiation
1.  Composed of perpendicular oscillating waves,
one for the ________ and one for the _________
field
1
B.  All electromagnetic radiation waves move through
space at the same, constant speed:
2
C.  Wave Characteristics:
1.  Wavelength (λ): distance from peak to peak
2.  Amplitude: magnitude or intensity of wave
3
3. 
Frequency (ν) is the number of waves that pass
a point in a given period of time
4. 
The total energy is proportional to the amplitude
of the waves and the frequency
a.  Larger amplitude =
b.  Higher frequency =
4
5. 
Wavelength and Frequency are related by the
speed of light:
α.  ν in s-1, λ in meters!
b.  Higher frequency = higher energy
c.  Higher wavelength = lower energy
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D.  Electromagnetic Spectrum
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E.  Wave Interference
1.  The interaction between waves is called
____________.
2.  When waves interact so that they add to make a
larger wave it is called ___________interference.
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4. 
When waves interact so they cancel each other it
is called __________ interference.
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F. 
Wave Diffraction
1.  When traveling waves encounter an obstacle or
opening in a barrier that is about the same size
as the wavelength, they bend around it – this is
called __________.
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2. 
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3. 
The diffraction of light through two slits separated
by a distance comparable to the wavelength
results in an _________________ of the
diffracted waves.
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G.  Ex: The relationship between ν and λ: The longest
wavelength of red light is λ = 750 nm. What is its
frequency?
H.  Ex: For the 3G cell phone network, ν = 3 GHz.
What is the wavelength of this electromagnetic
radiation?
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II. 
The Particle-like Nature of Light
A.  Light (and all electromagnetic radiation) exists as
discrete packets called photons, each with an
energy:
1. 
Ex: For the 3G cell phone network, ν = 3 GHz.
What is the energy of this electromagnetic
radiation?
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2. 
Light is quantized : only available in discrete,
whole numbers of photons.
A. 
B.  Ex: A 100 watt light bulb radiates 100 J per
second. If we assume that λ = 525 nm only,
how many photons are emitted each second?
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III.  Atomic Spectra and the Bohr Atom
A.  Sunlight is white light : contains a continous
spectrum of all visible colors (all λ), some infrared
and some ultraviolet too.
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B.  How do rainbows form?
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C.  Emission of light by atoms:
1.  Flame tests for aqueous metal ions.
Na
K
Li
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2. 
Gas discharge tubes.
Hg
He
H2
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Ba
3. 
Analyzing the light given off by H2 gas in a
discharge tube.
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4. 
Analyzing other gases.
H2
He
O2
Ne
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5. 
Atoms can also absorb light for an
absorption spectrum:
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5. 
Emission and Absorption spectra for
hydrogen, H2(g).
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D.  The Bohr Model of the Atom:
1.  Bohr s major idea was that the energy states
of the atom were _________, and that the
amount of energy in the atom was related to
the electron s position in the atom.
2.  The electrons travel in orbits that are at a
fixed distance from the nucleus.
3. 
Electrons emit radiation when they jump
from an orbit with higher energy down to an
orbit with lower energy
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E.  A picture of the Bohr Model for Hydrogen Atoms:
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F. 
For the hydrogen atom, the Rydberg equation
relates λ to the n values:
% 1 1(
1
= 1.097 ×107 m−1'' 2 − 2 **
λ
& n2 n1 )
1. 
€
2. 
A negative sign on λ can mean that the light is
being emitted.
Ex: What is the wavelength of light emitted as
an electron jumps from n = 4 to n = 1?
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3. 
From λ and the Rydberg equation, the energy
of an electron in a hydrogen atom can be
calculated:
hc
E=
λ
€
%
1
1(
7 −1 1
= 1.097 ×10 m '' 2 − 2 **
λ
& n2 n1 )
€
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4. 
Ex: Draw a ladder diagram of energy levels
for the n = 1, 3 and ∞ energy levels in kJ/mol.
Define n = ∞ as the zero energy state.
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F. 
The Wave-like Nature of Matter
1. 
For an electron, (m = 9.1x10–31 kg), the wavelike nature of matter is significant…
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2. 
The deBroglie wavelength of matter is:
3. 
Ex: What is the wavelength of an electron
traveling at 2.2x106 m/s, the velocity of an
electron in the hydrogen atom?
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4. 
Ex: All matter has wave-like properties. What
is my wavelength as I walk across the room at
1 m/s?
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G.  When you try to observe the wave nature of the
electron, you cannot observe its particle nature –
and vice-versa.
1.  wave nature =
2.  particle nature =
H.  The wave and particle nature of the electron are
complementary properties.
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I. 
Heisenberg Uncertainty Principle: the product of
the uncertainties in both the position and speed
of a particle was inversely proportional to its
mass.
1. 
2. 
3. 
4. 
x = position,
v = velocity,
m = mass in kg
h = Planck s constant
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5. 
Ex: An electron traveling at 2.2x106 m/s has
an uncertainty of 2x105 m/s in its velocity.
What is the minimum uncertainty in its
position?
(Δx)(mΔv) ≥
h
4π
€
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6. 
For electrons, the uncertainty in its position
and/or velocity prevent exact knowledge of its
future position.
7. 
Exact knowledge of the electrons position
collapses the diffraction pattern.
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IV.  Quantum Mechanics tells us about the location of the
electrons in the atom.
A.  Schrödinger s Equation allows us to calculate the
probability of finding an electron with a particular
amount of energy at a particular location in the atom.
B.  The solutions to Schrödinger s Equation give 4
quantum numbers that define orbitals that the
electrons are in. We look at the first 3 quantum
numbers this chapter. The last is in Ch. 8.
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1. 
The principal quantum number, n, describes
the main energy level (or shell) that an electron
occupies.
a.  The principal energy levels are very similar to
the Bohr Energy levels.
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2. 
The angular momentum quantum number, l,
describes the shape of the orbital. Each value of l
has specific a specific shape.
3. 
The magnetic quantum number, ml, designates
the orientation of the orbital within the sublevel.
4. 
The first principal energy level:
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5. 
The second principal energy level:
n = 2, l = 0 to 1
for l = 0, ml = 0
for l = 1, ml = –1, 0, 1
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6. 
The third principal energy level:
n = 3, l = 0 to 2
for l = 0, ml = 0
for l = 1, ml = –1, 0, 1
for l = 2, ml = –2, –1, 0, 1, 2
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7. 
The fourth principal energy level:
n = 4, l = 0 to 3
for l = 0, ml = 0
for l = 1, ml = –1, 0, 1
for l = 2, ml = –2, –1, 0, 1, 2
for l = 3, ml = –3, –2, –1, 0, 1, 2, 3
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8. 
The relationship between 1s, 2s and 3s.
9. 
The relationship between 2s and 2p.
10.  p and d orbitals have lobes with different
phases.
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D.  How to get from ψ to orbitals
1.  ψ is a function that is a solutionto the
Schrödinger equation. It is called a wavefunction.
2.  ψ2 is the probability density:
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3. 
The radial distribution function represents the
total probability at a certain distance from the
nucleus. It multiplies ψ2 times the volume at a
given distance r from the nucleus.
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4. 
The 2s and 3s orbitals have nodes.
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E.  Describing an Orbital
1.  Each set of n, l, and ml describes one orbital
2.  Orbitals with the same value of n are in the
same principal energy level.
3.  Orbitals with the same values of n and l are
said to be in the same sublevel.
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4. 
Ex: What are the quantum numbers and
names (for example, 2s, 2p) of the orbitals in
the n = 4 principal level? How many orbitals
exist?
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E.  Using Microsoft Excel to Plot ψ, ψ2 and ΔV*ψ2
1. 
ψ = wavefunction
1s: psi vs. radius
0.0016
0.0014
0.0012
psi
0.001
0.0008
0.0006
0.0004
0.0002
0
0
50
100
150
200
250
300
350
400
450
radius (pm)
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2. 
ψ2 = probability density of finding the electron at a
certain distance from the nucleus:
1s: psi^2 vs. radius
0.0016
0.0014
0.0012
psi^2
0.001
0.0008
0.0006
0.0004
0.0002
0
0
50
100
150
200
250
radius (pm)
48
300
350
400
450
ΔV*ψ2 = total radial probability at a given radius
= radial distribution function
3. 
1s: Total Radial Probability vs. Radius
Total Radial Probability
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
50
100
150
Radius (pm)
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1s: Total Radial Probability vs. Radius
Total Radial Probability
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
50
100
150
200
250
300
Radius (pm)
50
200
250
300