Chapter 7 : The Quantum-Mechanical Model of the Atom

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Chapter 7 The Quantum-Mechanical Model of the
Atom

• Outline
• Intro to Quantum Mechanics
• The Nature of Light
• Atomic Spectroscopy and the Bohr Model
• The Wave of Nature of Matter
• Quantum Mechanics and the Atom
• The Shapes of Atomic Orbitals

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The Behavior of the Very Small

• electrons are incredibly small

• electron behavior determines much of the behavior
  of atoms

• directly observing electrons in the atom is
  impossible

Intro to QM
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A Theory that Explains Electron Behavior

• the quantum-mechanical model explains the manner
  electrons exist and behave in atoms

• helps us understand and predict the properties of
  atoms that are directly related to the behavior
  of the electrons

• why some elements are metals while others are
  nonmetals
• why some elements are very reactive while others
  are practically inert

Intro to QM
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The Nature of Light
Before we introduce quantum mechanics we must
first understand a few things about light.

• light is a form of electromagnetic radiation
• composed of perpendicular oscillating waves, one
  for the electric field and one for the magnetic
  field

• all electromagnetic waves move through space at
  the same constant speed

Light
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Electromagnetic Radiation
EM radiation can be described as a wave composed
of oscillating electric and magnetic fields.
Light
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Waves
What do we mean by waves ?
Light
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Characterizing Waves
Maximum height above centre line (or the maximum
depth below the centre line) is the amplitude
Distance between successive peaks is called the
wavelength (?, lambda)
Number of peaks (or troughs) that pass through a
given point in a unit of time is the frequency,
(?, nu).
The difference in time between successive
occurrences of the same displacement is the
period. (t)
Light
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Relating Wavelength and Frequency

• for waves traveling at the same speed, the
  shorter the wavelength, the more frequently they
  pass

• this means that the wavelength and frequency of
  electromagnetic waves are inversely proportional

• since the speed of light is constant, if we know
  wavelength we can find the frequency, and vice
  versa

Light
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Wavelength x Frequency speed of light
Speed of light is known and equal to 3.00 x 108
m s-1
This equation applies to all the forms of
electromagnetic radiation (not just visible
light).
Light
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Types of Electromagnetic Radiation
There are many forms of EM radiation that you may
already be familiar with
Light
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Light
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A Closer Look at Visible Light
The colour of visible light depends on its
wavelength.
Visible light wavelengths are on the order of
100s of nm.
Light
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Example Questions
1. Many cordless phones operate on signals at 600
MHz. What is the equivalent wavelength ?
2. HeliumNeon lasers (the light used to scan
your groceries at the checkout) produce light at
633 nm. What is the frequency of the lasers
light ?
Light
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Properties of EM Radiation
Interference
When two sets of waves (for example water waves)
intersect, there are places where the waves
disappear and other places where the waves
persist.
Light
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When the waves are in-step, (called being
in-phase) the waves add together to give the
highest crests and the deepest troughs.
When the waves are out-of-step, (called being
out-of-phase) the waves cancel each other out.
Light
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Properties of EM Radiation
Diffraction
When a wave encounters an obstacle or a slit that
is comparable in size to its wavelength, it bends
around. This phenomenon is called diffraction.
Light
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Light
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So Light is a Wave
The Photoelectric Effect
When light strikes the surface of certain metals,
electrons are detected. This was first observed
by Heinrich Hertz in 1888 (12 years before
Plancks quantum theory).

• Electron emission only occurs when the _________
  of the light exceeds a threshold value.
• The number of electrons emitted depends on the
  ________ of the light but
• The kinetic energy of the emitted electrons
  depends on the __________of the light.

Light
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Light
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Einstein Explains the PEE

• Einstein proposed that the light energy was
  delivered to the atoms in packets, called quanta
  or photons

• the energy of a photon of light was directly
  proportional to its frequency
• __________ proportional to it wavelength
• the proportionality constant is called Plancks
  Constant, (h) and has the value 6.626 x 10-34 Js

Light
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• 1 photon at the threshold frequency has just
  enough energy for an electron to escape the atom

• for higher frequencies, the electron absorbs more
  energy than is necessary to escape

• this excess energy becomes kinetic energy of the
  ejected electron

Light
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Atomic Spectra
The atoms of group 1 give a characteristic colour
when placed in a flame.
Atomic Spectroscopy
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Exciting Gas Atoms to Emit Light with Electrical
Energy
Atomic Spectroscopy
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Emission Spectra
Atomic Spectroscopy
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The Atomic Spectrum of Hydrogen
Atomic Spectroscopy
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Example Question

1. Use the Rydberg equation for n 3. Does it agree
   with the experimental atomic spectra for hydrogen
   ?
2. Repeat the exercise for n 7. Can you explain
   why this line is not observed by the human eye ?

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Bohrs Model

• the electrons traveled in orbits that were a
  fixed distance from the nucleus
• therefore the _______of the electron was
  proportional to the distance the orbital was from
  the nucleus

• Niels Bohr proposed that the electrons could only
  have very specific amounts of energy

• electrons emitted radiation when they jumped
  from an orbit with higher energy down to an orbit
  with lower energy

Bohrs Model of the Atom
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Bohrs Model of the Atom
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The Wave Nature of Matter
if electrons behave like particles, there should
only be two bright spots on the target
Wave Nature of Matter
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• de Broglie proposed that ____particles could have
  wave-like character
• Incredibly, electrons which we were thought of as
  negatively charged _______also exhibit ________
  properties
• because it is so small, the wave character of
  electrons is significant
• de Broglie predicted that the wavelength of a
  particle was _________ proportional to its
  momentum

Wave Nature of Matter
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Examples
1. Calculate the de Broglie length of an electron
travelling at one-tenth the speed of light.
2. In last nights ALCS, a fastball was clocked
at 97 miles an hour (43 m/s). Given that a
baseball weighs 145 g. Calculate the de Broglie
length of his fastball and comment on whether
that was a feasible reason why the batters
couldnt hit the pitches.
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Uncertainty Principle

• Heisenberg stated that the product of the
  uncertainties in both the position and speed of a
  particle was inversely proportional to its mass
• x position,
• v velocity,
• m mass
• the means that the more accurately you know the
  position of a small particle, like an electron,
  the less you know about its speed
• and vice-versa

Wave Nature of Matter
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Quantum Mechanics
Standing waves are waves where the magnitude of
the oscillation is different from point to point
along the wave. Points that undergo no
displacement are called nodes.
Consider a plucked guitar string of length, l.
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n Number of nodes
1
2
3
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Particle in a Box (PIAB)
Schrödinger suggested that if an electron in an
atom has wave-like properties then it should be
describable using a mathematical equation called
a wavefunction (?). The wavefunction must be a
solution to Schrödingers equation The
wavefunction should correspond to a standing wave
within the boundary of the system being
described..
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1. The energy of the particle in a 1D PIAB is
quantized.

1. The minimum energy of the particle in a 1D PIAB
   is never zero

3. n is called the __________ quantum number
4. The square of the wavefunction, ?2, at a given
point in space represents the __________ of
finding the particle there.
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Exercise
Use de Broglies equation for matter waves, the
fact that the kinetic energy of a particle is
given by the following expression
and the equation for the wavelength of a standing
wave
to derive the equation for the energy of a 1D
PIAB.
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Quantum Mechanics

• The energy of an electron dictates the properties
  of an element. For example, bonding.

• However, if we very accurately know the energy of
  an electron, Heisenberg says we cant precisely
  know its position.

• for an electron with a given energy, the best we
  can do is describe a region in the atom of high
  probability of finding it

• To determine the energy of an electron the
  Schrödinger equation must be solved.

Quantum Mechanics
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Wave Function, y

• A wavefunction, ?, is just a mathematical
  function that is a solution to the Schrödinger
  equation.

• The square of the wavefunction, ?2 gives a
  probability map of finding the electron in a
  region of space.

• calculations show that the size, shape and
  orientation in space of an orbital are determined
  be three integer terms in the wave function

• these integers are called quantum numbers
• __________quantum number, n
• __________ momentum quantum number, l
• __________ quantum number, ml

Quantum Mechanics
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Principal Quantum Number, n

• characterizes the energy of the electron in a
  particular orbital

• n can be any integer ³ 1
• the larger the value of n,

• energies are defined as being negative
• the larger the value of n, the larger the orbital

Quantum Mechanics
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Angular Momentum Quantum Number, l

• The angular quantum number is an integer that
  determines the shape of the orbital (see later).

• Possible values for l are 0,1,2,,(n-1).

Value of l Letter Designation
0
1
2
3
Quantum Mechanics
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Magnetic Quantum Number, ml

• The magnetic quantum number is an integer that
  determines the orientation of the orbital (see
  later).

• Possible values for ml are l, (l-1), (l-2)-l.

• Each specific combination of n,l,ml specifies one
  atomic orbital.

• Orbitals with the same principal quantum number
  are said to be in the same principal level
  (shell).

• Orbitals with the same value of n and m are said
  to be in the same sublevel (subshell).

Quantum Mechanics
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Example
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Levels and Sublevels
Quantum Mechanics
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Orbital energies for a hydrogen atom depend only
on the principal quantum number n. This means
that all the subshells within a principal shell
have the same energy. Orbitals at the same energy
level are said to be __________.
Electronic Orbitals of Hydrogen
Quantum Mechanics
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Example
Question 32 from Tro (Chapter 7 End of Chapter
Problems)
List all the orbitals in each of the following
principal levels. Specify the three quantum
numbers for each orbital.

1. n 1
2. n 2
3. n 3
4. n 4

Quantum Mechanics
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Principal Energy Levels in Hydrogen
Quantum Mechanics
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The Hydrogen Spectrum Explained !

• both the Bohr and Quantum Mechanical Models can
  predict these lines very accurately

Quantum Mechanics
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Quantum Mechanics
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The Shapes of Atomic Orbitals
Recall that ?2 gives the probability density
The Shapes of Atomic Orbitals
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The Radial Distribution Function
Function Meaning
Prob. density Probability of finding e- at a _______ r
RDF Probability of finding e- at a _______r
The Shapes of Atomic Orbitals
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A node is a point where both ? and ?2 all equal
zero.
The ns orbitals (n gt 1) are spherically symmetric
like the 1s orbital. They are just bigger and
have nodes.
The Shapes of Atomic Orbitals
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The Shapes of Atomic Orbitals
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p Orbitals (l 1)
There are three types of p orbitals. Each
corresponds to a different ml quantum number.
The Shapes of Atomic Orbitals
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d Orbitals (l 2)
The Shapes of Atomic Orbitals
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Example
Write an orbital designation corresponding to the
quantum numbers n 4, l 2, ml 0.
Write an orbital designation corresponding to the
quantum numbers n 3, l 1, ml 1.