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1
George Mason University General Chemistry
211 Chapter 7 Quantum Theory and Atomic
Structure Acknowledgements Course Text
Chemistry the Molecular Nature of Matter and
Change, 6th edition, 2011, Martin S. Silberberg,
McGraw-Hill The Chemistry 211/212 General
Chemistry courses taught at George Mason are
intended for those students enrolled in a science
/engineering oriented curricula, with particular
emphasis on chemistry, biochemistry, and biology
The material on these slides is taken primarily
from the course text but the instructor has
modified, condensed, or otherwise reorganized
selected material.Additional material from other
sources may also be included. Interpretation of
course material to clarify concepts and solutions
to problems is the sole responsibility of this
instructor.
2
Quantum Theory of The Atom
  • How are electrons distributed in space?
  • What are electrons doing in the atom?
  • The nature of the chemical bond must first be
    approached by a closer examination of the
    electrons
  • Electrons are involved in the formation of
    chemical bonds between atoms
  • Quantum theory explains more about the electronic
    structure of atoms

3
Quantum Theory of The Atom
  • Origin of Atomic Theory
  • When burned in a flame metals produce colors
    characteristic of the metal.
  • This process can be traced to the behavior of
    electrons in the atom.

4
Quantum Theory of The Atom
  • Emission (line) Spectra of Some Elements
  • When elements are heated in a flame and their
    emissions passed through a prism, only a few
    color lines exist and are characteristic for each
    element. Atoms emit light of characteristic
    wavelengths when excited (heated)

5
Quantum Theory of The Atom
  • Electrons and Light Wave Nature of Light
  • Light moves (propagates) along as a wave (similar
    to ripples from a stone thrown in water)
  • Light consists of oscillations of electric and
    magnetic fields that travel through space
  • All electromagnetic radiation
  • Visible light, Microwaves, Radio Waves,
    Ultraviolet Light, X-rays, Infrared Light
  • consists of energy propagated by means of
    electric and magnetic fields that alternately
    increase and decrease in intensity as they move
    through space

6
Quantum Theory of The Atom
  • A Light Wave is Propagated as an Oscillating
    Electric Field (Energy)
  • The Wave properties of electromagnetic radiation
    are described by two independent variables
    wavelength and frequency

? wavelength, equiv. crest to crest distance ?
c/? frequency, cycles per second (Hz) c
speed of electromagnetic radiation (3 x 108 m/s)
7
Quantum Theory of The Atom
  • Wave Nature of Light
  • Wavelength the distance between any two
    adjacent identical points in a wave (given the
    notation ?
  • Frequency number of wavelengths that pass a
    fixed point in one unit of time (usually per
    second, given the notation ?). The common unit
    of freq is hertz (Hz, /s)
  • Propagation (Velocity) of an Electromagnetic wave
    is given as
  • c ??
  • c velocity 3.0 x 108 m/s in a vacuum
  • c is independent of ? or ? in a vacuum

8
Quantum Theory of The Atom
  • Relationship Between Wavelength and Frequency
  • Wavelength and frequency (c ??) are inversely
    proportional

9
The electromagnetic spectrum
Frequency (?)
High
Low
Energy (E)
High
Low
Wavelength (?)
Short
Long
10
Distinction between Energy Matter
  • At the macro scale level everyday life energy
    matter behave differently

11
Distinction between Energy Matter
12
Distinction between Energy Matter
  • The diffraction pattern caused by waves passing
    through two adjacent slits
  • Constructive and destructive interference occurs
    as water waves viewed from above pass through two
    adjacent slits in a ripple tank
  • As light waves pass through two closely spaced
    slits, they also emerge as circular waves and
    interfere with each other
  • They create a diffraction (interference) pattern
    of bright regions where crests coincide in phase
    and dark regions where crests meet troughs (out
    of phase) cancelling each other out

13
Quantum Theory of The Atom
  • Quantum Effects Wave-Particle Duality
  • Wave-Particle Duality is a central concept in
    Chemistry Physics
  • All matter and energy exhibits both wave-like and
    particle-like properties
  • Duality applies to macroscopic (large scale)
    objects, microscopic objects (atoms and
    molecules), and quantum objects (elementary
    particles protons, neutrons, quarks, mesons)
  • As atomic theory evolved, matter was generally
    thought to consist of particles
  • At the same time, light was thought to be a wave

14
Quantum Theory of The Atom
  • Quantum Effects - Wave-Particle Duality
  • Christiaan Huygens proposed the wave theory of
    light
  • Huygens wave theory was displaced by Isaac
    Newtons view that light consisted of a beam of
    particles
  • In the early 1800s Young and Fresnel showed that
    light, like waves, could be diffracted and
    produce interference patterns, confirming
    Huygens view
  • In the late 1800s James Maxwell developed
    equations, later verified by experiment, that
    explained light as a propagation of
    electromagnetic waves
  • At the turn of the 20th century, physicists began
    to focus on 3 confounding phenomena to explain
    Wave-Particle Duality
  • Black Body radiation
  • The Photoelectric Effect
  • Atomic Spectra

15
Quantum Theory of The Atom
  • Quantum Effects Wave-Particle Duality
  • Black Body Radiation - As the temperature of an
    object changes, the intensity and wavelength of
    the emitted light from the object changes in a
    manner characteristic of the idealized
    Blackbody in which the temperature of the body
    is directly related to the wavelengths of the
    light that it emits
  • In 1901, Max Planck developed a mathematical
    model that reproduced the spectrum of light
    emitted by glowing objects
  • His model had to make a radical assumption (at
    that time)
  • A given vibrating (oscillating) atom can have
    only certain quantities of energy and in turn can
    only emit orabsorb only certain quantities of
    energy

16
Quantum Theory of The Atom
  • Quantum Effects Wave-Particle Duality
  • Plancks Model
  • E (Energy of Radiation)
  • v (Frequency)
  • n (Quantum Number) 1,2,3
  • h (Plancks Constant, a Proportionality
    Constant)
  • 6.626 x 10-34 J ? s)
  • 6.626 x 10-34 kg ? m2/s
  • Atoms, therefore, emit only certain quantities of
    energy and the energy of an atom is described as
    being quantized
  • Thus, an atom changes its energy state by
    emitting (or absorbing) one or more quanta

17
Quantum Theory of The Atom
  • Wave-Particle Duality The Photoelectric
    Effect
  • The Planck model views emitted energy as waves
  • Wave theory associates the energy of the light
    with the amplitude (intensity) of the wave, not
    the frequency (color)
  • Wave theory predicts that an electron would break
    free of the metal when it absorbed enough energy
    from light of any color (frequency)
  • Wave theory would also imply a time lag in the
    flow of electric current after absorption of the
    radiation
  • Both of these observations are at odds with the
  • Photoelectric Effect

18
Quantum Theory of The Atom
  • Wave-Particle Duality The Photoelectric
    Effect
  • Photoelectric Effect
  • Flow of electric current when monochromatic light
    of sufficient frequency shines on a metal plate
  • Electrons are ejected from the metal surface,
    only when the frequency exceeds a certain
    threshold characteristic of the metal.
  • Radiation of lower frequency would not produce
    any current flow no matter how intense
  • Violet light will cause potassium to eject
    electrons, but no amount of red light (lower
    frequency) has any effect
  • Current flows immediately upon absorption of
    radiation

19
Quantum Theory of The Atom
  • Wave-Particle Duality The Photoelectric
    Effect
  • Einstein resolved these discrepancies
  • He reasoned that if a vibrating atom changed
    energy from nhv to (n-1)hv, this energy would be
    emitted as a quantum (hv) of light energy he
    called a photon
  • He defined the photon as a Particle of
    Electromagnetic energy, with energy E,
    proportional to the observed frequency of the
    light.
  • The energy (hv) of an impacting photon is taken
    up (absorbed) by the electron and ceases to exist
  • The Wave-Particle Duality of light is regarded as
    complimentary views of wave and particle pictures
    of light

20
Quantum Theory of The Atom
  • In 1921 Albert Einstein received the Nobel Prize
    in Physics for discovering the photoelectric
    effect
  • Electrons in metals exist in different and
    specific energy states
  • Photons whose frequency matches or exceeds the
    energy state of the electron will be absorbed
  • If the photon energy (frequency) is less than the
    electron energy level, the photon is not absorbed
  • The electron moves to a higher energy state and
    is ejected from the surface of the metal
  • The electrons are attracted to the positive anode
    of a battery, causing a flow of current

21
Practice Problem
  • Light with a wavelength of 478 nm lies in the
    blue region of the visible spectrum.
  • Calculate the frequency of this light
  • Speed of Light 3 x 108 m/s
  • Ans

22
Practice Problem
  • The green line in the atomic spectrum of Thallium
    (Tl) has a wavelength of 535 nm.
  • Calculate the energy of a photon of this light?

23
Practice Problem
  • At its closest approach, Mars is 56 million km
    from earth. How many minutes would it take to
    send a radio message from a space probe of Mars
    to Earth when the planets are at this closest
    distance?

24
Quantum Theory of The Atom
  • Atomic Line Spectra
  • When light from excited (heated) Hydrogen atoms
    or other atoms passes through a prism, it does
    not form a continuous spectrum, but rather a
    series of colored lines (Line Spectra) separated
    by black spaces
  • The wavelengths of these lines are characteristic
    of the elements producing them
  • The spectra lines of Hydrogen occur in several
    series, each series represented by a positive
    integer, n

25
Quantum Theory of The Atom
n 1
n 2
n 3
26
Quantum Theory of The Atom
  • Atomic Line Spectra
  • In 1885, J. J. Balmer showed that the
    wavelengths, l, in the visible spectrum of
    Hydrogen could be reproduced by a Rydberg
    Equation
  • where R The Rydberg Constant
  • ? wavelength of the spectral
    line
  • n1 n2 are positive integers and
    n2 gt n1

27
Quantum Theory of The Atom
  • Atomic Line Spectra
  • For the visible series of lines the value of n1
    2
  • The known wavelengths of the four visible lines
    for hydrogen correspond to values of n2 3, n
    4,n 5, and n 6
  • The Rydberg equation becomes
  • The above equation and the value of R are based
    on data rather than theory
  • The following work of Niels Bohr makes the
    connection between the data model and Theory

with n2 3, 4, 5, 6.
28
Quantum Theory of The Atom
  • Bohr Theory of the Hydrogen Atom
  • Prior to the work of Niels Bohr, the stability of
    the atom could not be explained using the
    then-current theories
  • How can electrons (e-) lose energy and remain in
    orbit?
  • Bohr in 1913 set down postulates to account for
    (1) the stability of the hydrogen atom and (2)
    the line spectrum of the atom
  • Energy level postulateAn electron can have only
    specific energy levels in an atom
  • Transitions between energy levelsAn electron in
    an atom can change energy levels by undergoing a
    transition from one energy level to another

29
Quantum Theory of The Atom
  • Bohr Theory
  • Transitions of the Electron in theHydrogen Atom

30
Quantum Theory of The Atom
  • Bohr Theory of the Hydrogen Atom
  • Bohrs Postulates
  • Bohr derived the following formula for the energy
    levels of the electron in the hydrogen atom

31
Quantum Theory of The At
  • Bohr Theory of the Hydrogen Atom
  • Bohrs Postulates
  • For the Hydrogen atom, Z 1
  • For the energy of the ground state (n 1)

32
Quantum Theory of The Atom
  • Bohr Theory of the Hydrogen Atom
  • Bohrs Postulates
  • When an electron undergoes a transition from a
    higher energy level (ni) to a lower one (nf), the
    energy is emitted as a photon

33
Quantum Theory of The Atom
  • Bohr Theory of the Hydrogen Atom
  • Bohrs Theory vs. Rydberg Data model
  • If we make a substitution into the previous
    equation that states the energy of the emitted
    photon, h?, equals hc/?
  • Thus, from the classical relationships of charge
    and motion combined with the concept of discreet
    energy levels theory matches data

versus
Bohr (theory)
Rydberg (data)
34
Practice Problem
  • From the Bohr model of the hydrogen atom we can
    conclude that the energy required to excite an
    electron from n 2 to n 3 is ___________ the
    energy to excite an electron from n 3 to n 4
  • a. less than b. greater than
  • c. equal to d. either equal to or less than
  • e. either equal to or greater than

Greater Than
35
Practice Problem
  • An electron in a hydrogen atom in the level n 5
    undergoes a transition to level n 3. What is
    the wavelength of the emitted radiation?
  • (R 2.179 x 10-18 J)

Note For computation of frequency and
wavelength the negative sign of the
energy value can be ignored
36
Quantum Theory of The Atom
  • Bohr Theory of the Hydrogen Atom
  • Bohrs Postulates
  • Bohrs theory explains not only the emission of
    light, but also the absorption of light
  • When an electron falls from n 3 to n 2 energy
    level, a photon of red light (wavelength, 685 nm)
    is emitted
  • When red light of this same wavelength shines on
    a hydrogen atom in the n 2 level, the energy is
    gained by the electron that undergoes a
    transition to n 3

37
Quantum Theory of The Atom
  • Quantum Mechanics
  • Bohrs theory established the concept of atomic
    energy levels but did not thoroughly explain the
    wave-like behavior of the electron
  • Current ideas about atomic structure depend on
    the principles of quantum mechanics, a theory
    that applies to subatomic particles such as
    electrons. Electrons show properties of both
    waves and particles

38
Quantum Theory of The Atom
  • Quantum Mechanics
  • The first clue in the development of quantum
    theory came with the discovery of thede Broglie
    relation
  • In 1923, Louis de Broglie reasoned that if light
    exhibits particle aspects, perhaps particles of
    matter show characteristics of waves
  • He postulated that a particle with mass m and a
    velocity v has an associated wavelength
  • The equation ? h/mv is called the
  • de Broglie relation

39
Quantum Theory of The Atom
  • Quantum Mechanics
  • If matter has wave properties, why are they not
    commonly observed?
  • The de Broglie relation shows that a baseball
    (0.145 kg) moving at about 60 mph (27 m/s) has a
    wavelength of about 1.7 x 10-34 m.
  • This value is so incredibly small that such waves
    cannot be detected.
  • Electrons have wavelengths on the order of a few
    picometers (1 pm 10-12 m).

40
Practice Problem
  • At what speed must an neutron (1.67 x 10-27 kg)
    travel to have a wavelength of 10.0 pm?

41
Quantum Theory of The Atom
  • Quantum mechanics is the branch of physics that
    mathematically describes the wave properties of
    submicroscopic particles
  • We can no longer think of an electron as having a
    precise orbit in an atom
  • To describe such an orbit would require knowing
    its exact position and velocity, i.e., its motion
    (mv)
  • In 1927, Werner Heisenberg showed (from quantum
    mechanics) that it is impossible to
    simultaneously measure the present position while
    also determining the future motion of a particle,
    or of any system small enough to require quantum
    mechanical treatment

42
Quantum Theory of The Atom
  • Max Born stated in his Nobel Laureate speech
  • To measure space coordinates and instants of
    time, rigid measuring rods and clocks are
    required.
  • On the other hand, to measure momenta and
    energies, devices are necessary with movable
    parts to absorb the impact of the test object and
    to indicate the size of its momentum (mass x
    velocity).
  • Paying regard to the fact that quantum mechanics
    is competent for dealing with the interaction of
    object and apparatus, it is seen that no
    arrangement is possible that will fulfill both
    requirements simultaneously.

43
Quantum Theory of The Atom
  • Mathematically, the uncertainty relation between
    position and momentum, i.e., the variables,
    arises due to the fact that the expressions of
    the wavefunction in the two corresponding bases
    (variables) are Fourier transforms of one
    another.
  • According to the uncertainty principle of
    Heisenberg, if the two operators representing a
    pair of variables do not commute, then that pair
    of variables are mutually complementary, which
    means they cannot be simultaneously measured or
    known precisely.
  • In the mathematical formulation of quantum
    mechanics, changing the order of the operators
    changes the end result, i.e., the operators are
    non-commuting, and are subject to similar
    uncertainty limits.

44
Quantum Theory of The Atom
  • Quantum Mechanics
  • Heisenbergs uncertainty principle is a relation
    that states that the product of the uncertainty
    in position (Dx) and the uncertainty in momentum
    (mDvx) of a particle can be no smaller than
  • When m is large (for example, a baseball) the
    uncertainties are very small, but for electrons,
    high uncertainties disallow defining an exact
    orbit

45
Practice Problem
  • Heisenberg's uncertainty principle can be
    expressed mathematically as
  • Where ?x is the uncertainty in position
  • ?p ( m?v) is the uncertainty in Momentum
  • h is Planck's constant (6.626 x 10-34 kg ? m2/s)
  • What would be the uncertainty in the position
    (?x) of a fly (mass 1.245 g) that was traveling
    at a velocity of 3.024 m/s if the velocity has an
    uncertainty of 2.72?

Plancks Constant h 6.626 x 10-34 J ? s 1 J 1
kg ? m2/s2 h 6.626 x 10-34 kg ? m2/s
46
Quantum Theory of The Atom
  • Quantum Mechanics
  • Acceptance of the dual nature of matter and
    energy (E mc2) and the Uncertainty Principle
    culminated in the field of Quantum Mechanics
  • Wave Nature of objects on the Atomic Scale
  • Erwin Schrodinger developed quantum mechanical
    model of the Hydrogen atom, where
  • An Atom has certain allowed quantities of energy
  • An Electrons behavior is wavelike, but its exact
    location is impossible to know
  • The Electrons Matter-Wave occupies 3-dimentional
    space near nucleus
  • The Matter-Wave experiences continuous, but
    varying influence from the nuclear charge

47
Quantum Theory of The Atom
  • Quantum Mechanics
  • Schrodinger Equation
  • H? ??
  • ? Energy of the atom
  • ? Wave Function
  • H Hamiltonian Operator Mathematical
    operations that when carried out on a
    particular wave yields the allowed energy
    value
  • Each solution of the wave equation is associated
    with a given atomic orbital, which bears no
    resemblance to an orbit in the Bohr model
  • An Orbital is a mathematical function, which
    like a Bohr Orbit, represents a particular energy
    level of the orbiting electron, but it has no
    direct physical meaning

48
Quantum Theory of The Atom
  • Quantum Mechanics
  • Heisenberg's uncertainty principle says we cannot
    precisely define an electrons orbit
  • The wave function (atomic orbital) has no direct
    physical meaning
  • The square of the wave function, ? 2, however,
    is defined as the probability density, a measure
    of the probability that the electron can be found
    within a particular tiny volume of the atom

49
Quantum Theory of The Atom
  • Probability of Finding an Electron in a Spherical
    Shell About the Nucleus

50
Quantum Theory of The Atom
  • Quantum Numbers and Atomic Orbitals
  • According to quantum mechanics each electron is
    described by 4 quantum numbers
  • Principal Quantum Number (n)
  • Angular Momentum Quantum Number (l)
  • Magnetic Quantum Number (ml)
  • Spin Quantum Number (ms)
  • The first three quantum numbers define the wave
    function of the electrons atomic orbital
  • The fourth quantum number refers to the spin
    orientation of the 2 electrons that occupy an
    atomic orbital

51
Quantum Theory of The Atom
  • Quantum Numbers and Atomic Orbitals
  • The Principal Quantum Number (n) represents the
    Shell Number in which an electron resides
  • It represents the relative size of the orbital
  • Equivalent to periodic chart Period Number
  • Defines the principal energy of the electron
  • The smaller n is, the smaller the orbital
  • The smaller n is, the lower the energy of the
    electron
  • n can have any positive value from
  • 1, 2, 3, 4 ?
  • (Currently, n 7 is the maximum known)

52
Quantum Theory of The Atom
  • Quantum Numbers and Atomic Orbitals (Cont)
  • The Angular Momentum Quantum Number (l)
    distinguishes sub shells within a given shell
  • Each main shell, designated by quantum number
    n, is subdivided into
  • l n - 1 sub shells
  • (l) can have any integer value from 0 to n - 1
  • The different l values correspond to thes, p,
    d, f designations used in the electronic
    configuration of the elements
  • Letter s p d f
  • l value 0 1 2 3

53
Quantum Theory of The Atom
  • Quantum Numbers and Atomic Orbitals (Cont)
  • The Magnetic Quantum Number (ml) defines atomic
    orbitals within a given sub-shell
  • Each value of the angular momentum number (l)
    determines the number of atomic orbitals
  • For a given value of l, ml can have any
    integer value from -l to l
  • ml -l to l
  • Each orbital has a different shape and
    orientation (x, y, z) in space
  • Each orbital within a given angular momentum
    number sub shell (l) has the same energy

54
Quantum Theory of The Atom
  • Quantum Numbers and Atomic Orbitals (Cont)
  • The Spin Quantum Number (ms) refers to the two
    possible spin orientations of the electrons
    residing within a given atomic orbital
  • Each atomic orbital can hold only two (2)
    electrons
  • Each electron has a spin orientation value
  • The spin values must oppose one another
  • The possible values of ms spin values are
  • 1/2 and 1/2

55
Summary of Quantum Numbers
Name
Symbol
Permitted Values
Property
principal
n
positive integers (1, 2, 3, )
orbital energy (size)
angular momentum
l
integers from 0 to n -1
orbital shape The l values 0, 1, 2, and 3
correspond to s, p, d, and f orbitals,
respectively
magnetic
ml
integers from-l to 0 to l
orbital (x,y,z) orientation
spin
ms
1/2 or -1/2
e- spin orientation
56
Quantum Numbers and Atomic Orbitals
The Hierarchy of Quantum Numbers for Atomic
Orbitals
Note n gt 7 l gt 3 not defined for the current
list of elements in the Periodic Table
57
Quantum Numbers and Atomic Orbitals
  • Using calculated probabilities of electron
    position, the shapes of the orbitals can be
    described
  • The s (n 1) sub shell orbital (there is only
    one) is spherical
  • The p (n 2) sub shell orbitals (there are
    three) are dumbbell shape
  • The d (n 3) sub shell orbitals (there are
    five) are a mix of cloverleaf and dumbbell shapes

58
Quantum Numbers and Atomic Orbitals
  • Cross-sectional Representations of the
    Probability Distributions of s Orbitals
    (spherical)

59
Quantum Numbers and Atomic Orbitals
  • Cutaway Diagrams Showing the Spherical Shape of
    s Orbitals

60
Quantum Numbers and Atomic Orbitals
  • Radial Probability Distributionof the Three 2p
    Orbitals (dumbell shapes)

n 2 l 2 1 1 (p) ml -1
0 1
61
Quantum Numbers and Atomic Orbitals
  • Radial Probability Distribution
  • of the Five 3d Orbitals (Cloverleaf Dumbells)

n 3 l 3 1 2 (d) ml -2
-1 0 1 2
62
Quantum Numbers and Atomic Orbitals
  • Radial Probability Distribution
  • of the Seven 4f Orbitals

n 4 l 4 1 3 (f) ml -3 -2
-1 0 1 2 3
63
Quantum Numbers and Atomic Orbitals
  • Orbital Energies of the Hydrogen Atom

64
Practice Problems
  • If the n quantum number of an atomic orbital is
    4, what are the possible values of the l quantum
    number?
  • Ans (l) can have any integer value from 0 to n
    1
  • l n - 1 4 1 3
  • ? Values of l 0 1 2 3

65
Practice Problem
  • If the l quantum number is 3, what are the
    possible values of ml?
  • Ans ml can have any integer value from -l to
    l
  • Since l 3
  • ml -3 -2 -1 0 1 2 3

66
Practice Problem
  • State which of the following sets of quantum
    numbers would be possible and which impossible
    for an electron in an atom?
  • a. n 0, l0, ml 0, ms 1/2
  • b. n 1, l0, ml 0, ms 1/2
  • c. n 1, l0, ml 0, ms -1/2
  • d. n 2, l1, ml -2, ms 1/2
  • e. n 2, l1, ml -1, ms 1/2
  • Ans Possible
  • Impossible
  • Impossible

b c e
a n must be positive 1, 2, 3...
d ml can only be -1 0 1
67
Summary Equations
  • Light c ?? c 3 x
    108 m/s
  • Plancks Model
  • Photoelectric
  • Balmer Rydberg
  • Bohr Model
  • Bohr Postulate
  • De Broglie
  • Heisenberg
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